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SOLAR  AND  LUNAR 


ECLIPSES 


FAMILIARLY  ILLUSTRATED  AND  EXPLAINED, 


WITH   THE 


METHOD  OF  CALCULATING  THEM 


ACC0RDI5G   TO   THE 


THEORY  OF  ASTRONOMY, 


AS   TAUGHT  IH 


NEW    ENGLAND    COLLEGES 


BY  JAMES    H.    COFFIN,  A.M. 


NEW    YORK: 

COLLINS,    BROTHER    &    CO 
1845. 


Entered  according  to  Act  of  Congress,  in  the  year  1845, 

BY  JAMES  H.  COFFIN, 
In  the  Clerk's  Office  of  the  District  Court  of  Connecticut. 


541 
CC4 


PREFACE 


The  design  of  this  treatise,  is  to  explain  the  "  rationale"  of  some 
of  the  most  interesting  astronomical  calculations,  in  such  a  way 
that  the  student  may  clearly  see  the  reason  of  every  step,  and  its 
connection  with  the  theory.     In  this  respect  it  differs  from  many 
others,  which  give  the  rules  for  calculating  merely,  without  any 
explanation  of  the  reason  of  them.     Being  partly  designed  as  a  text 
hook  for  colleges,  the  author  has  endeavoured  to  adapt  it  to  the 
design  of  college  education,  which  is  not  so  much  to  make  adept 
practitioners  in  any  particular  science,  as  to  give  broad  and  com- 
prehensive views  of  the  whole  field.     Hence,  the  principles  of  the 
several  sciences  should  be  thoroughly  understood  by  the  student ; 
but  the  application  of  them  to  practice  by  mere  rules  is  foreign  to 
the  design  of  a  collegiate  course  of  study.     If,  therefore,  the  calcu- 
lations of  astronomy  are  attended  to  at  all  in  college,  it  should  be 
in  such  a  way,  that  the  connection  with  the  theory  may  be  appa- 
rent, and  that  the  two  may  mutually  illustrate  each  other.     In 
many  of  treatises  for  colleges,  this  point  seems  to  be  overlooked. 
Some  of  them  contain  tables  for  astronomical  calculations  which 
are  very  minute  and  accurate,  and  at  the  same  time,  so  constructed 
and  arranged  as  to  reduce  the  labour  of  calculation  as  much  as 
possible  ;  but  the  student  can  see  no  connection  between  them  and 
the  motions  and  perturbations  which  occupy  his  attention  in  the 
study  of  the  theory.     In  fact,  one  who  has  studied  the  theory  with 
ever  so  much  thoroughness,  has  here  very  little  advantage  over 
one  who  is  entirely  ignorant  of  it ;  each  being  guided  wholly  by 
rules  that  must  appear  entirely  arbitrary. 

Such  tables  not  being  adapted  to  the  design  of  this  treatise,  the 
author  found  it  necessary  to  prepare  a  set  differing  somewhat  in 

111505 


IV  PREFACE. 

their  construction  from  those  in  ordinary  use.  They  are  based, 
for  the  most  part,  on  those  of  Delambre  and  Burg,  with  the  more 
recent  improvements  of  Airy  and  Bessel ;  but  varied  in  the  plan  of 
construction  so  as  to  adapt  them  to  this  work.  Calculations  may 
be  made  from  them  sufficiently  accurate  for  the  ordinary  purposes 
of  an  almanac ;  yet,  as  they  are  not  designed  especially  for  that 
purpose,  the  aim  has  been  not  so  much  to  secure  extreme  accuracy 
in  the  results,  as  to  render  them  easily  understood. 

The  quantities  in  the  tables  are  given,  for  the  most  part,  in 
degrees  and  decimals,  instead  of  signs,  degrees,  minutes,  and  sec- 
onds, with  a  view  to  facilitate  the  labour  of  calculation,  and  to 
secure  the  same  degree  of  accuracy  with  a  less  number  of  figures. 

The  work  is  intended  to  be  complete  in  itself,  on  the  subject  on 
which  it  treats,  for  those  who  have  a  general  knowledge  of  the 
motions  of  the  heavenly  bodies  ;  yet,  it  would,  doubtless,  contribute 
to  a  better  understanding  of  it,  to  read  some  such  work  as  Olm- 
sted's or  Herschell's  Astronomy  previously. 

Williams  College,  October,  1843. 


CONTENTS. 


PART    I. 


CHAPTER  I. 

GENERAL  CONDITIONS  OF  AN   ECLIPSE. 
1,  Causes  of  eclipses.— 2.  Must  occur  near  the  Moon's  nodes — 3.  Moon's  latitude  in  solar 
eclipses. — 4.  Do.  in  lunar  eclipses. — 5.  Ecliptic  limits.— 6.  Solar. — 7.  Lunar. — 8.  Eclipses 
must  occur  six  months  apart. 

CHAPTER  II. 

MEAN  TIME  OF  AN  ECLIPSE. 
9.  Time  when  the  sun  passes  the  moon's  nodes,  how  found. — 10.  Motions  not  uniform. — 
11.  Mean  time  of  an  eclipse,  how  found. — 12.  Annual  equations  of  moon's  perigee  and 
node  explained. — 13.  The  same  applied. 

CHAPTER  ILL 

EQUATION  OF  THE  CENTRE. 
14.  Velocity  must  be  variable  in  eliptical  orbits.— 15.  The  same  shown  by  the  principle  of 
equal  areas  in  equal  times. — 15.  Equation  of  the  centre  explained. — 17.  The  same  applied. 

CHAPTER  IV, 


TIONS  OF  LONGITUDE. 
18.  Moon's  motion  disturbed  by  sun's  attraction. — 19.  Annual  equation  of  the  moon's  longi" 
tude  explained  and  applied. — 20.  Acceleration  of  the  moon's  mean  motion. — 21.  Its  discov- 
ery.— 22.  Theories  to  account  for  it. — 23.  Earth's  orbit  becoming  less  eliptical. — 24.  This 
shown  to  be  the  cause  of  the  acceleration. — 25.  Secular  equation  applied. 

CHAPTER  V. 

PERTURBATIONS  IN  THE  MOON's    MOTION,  CONTINUED. VARIATION. 

26.  Introduction  to  the  subject  of  the  chapter. — 27.  Variation. — 28.  Sun's  attraction  sometimes 
increases  and  sometimes  diminishes  moon's  gravity. — 29.  Moon's  motion  alternately  ac- 
celerated and  retarded. — 30.  Effect  upon  the  shape  of  the  orbit. — 31.  Combined  effects  of 
perturbation  in  velocity  and  shape. — 32.  The  Table  for  Variation  explained.— 33.  The 
equation  applied. — 34.  Annual  equation  of  variation  explained  and  applied. 

CHAPTER  VI. 

PERTURBATION  IN  THE  MOOn's  MOTION,  CONTINUED. EVECTION. 

35.  Difference  between  Variation  and  Evection. — 36.  Evection  caused,  in  part,  by  the  une- 
quable motion  of  the  apsides. — 37.  Caused,  in  part,  by  a  variation  in  the  eccentricity  of 
the  orbit. — 38.  Eccentricity  depends  on  relative  attraction. — 39.  The  principle  applied  to 
the  moon's  orbit. — 40.  Apsides  must  progress  on  the  whole. — 41.  Their  place,  how  deter- 
mined, physically. — 42.  Their  motion,  how  affected  by  an  increase  of  gravity. — 43.  How 


Vi  CONTENTS. 

by  a  diminution  of  the  same. — 44.  How  by  a  perturbation  in  velocity. — 45.  Neutral  points. 
— 46.  Addititious  force. — 47.  Ablatitious  force. — 48.  The  two  combined. — 47.  Neutral 
points  determined. — 50.  Increase  of  gravity  in  quadrature  only  equal  to  half  the  diminution 
in  syzygy. — 51.  Tangential  force. — 52.  Apsides  progress  when  within  45°  of  syzygy. — 
53.  Remain  nearly  stationary  when  between  45°  and  54°  43'  56". — 54.  Regress  when 
more  distant. — 55.  Progress  exceeds  the  regress. — 56.  Variation  in  eccentricity  and  motion 
of  apsides  represented  by  a  figure. — 57.  Evection  represented  by  the  same  figure. — 58. 
Argument  of  evection  explained. — 59.  The  same  computed. — 60.  Evectiun  applied. — 61. 
Annual  equation  of  evection. 

CHAPTER  VII. 

PERTURBATIONS  IN  THE  MOON'S  MOTION,  CONCLUDED. NODAL  EQUATION 

OF  THE  MOON'S  LONGITUDE. CORRECTION  OF  EQUATION  OF  THE  CEN- 
TRE.  REDUCTION  TO  THE  ECLIPTIC. 

62.  Disturbing  force  acts  out  of  the  plane  of  the  moon's  orbit.— 63.  Theory  of  nodal  equation 
explained.— 64.  Equation  applied. — 65.  Equation  of  the  centre  corrected.— 66.  Reduction 
explained  and  applied. 

CHAPTER  VIII. 

LUNAR  OR  MENSTRUAL  EQUATION  OF  THE  SUN*S  LONGITUDE. 
67.  Theory  of  lunar  equation.— 68.  The  same  applied.— 69.  Effect  of  inequalities  inappre- 
ciable. 

CHAPTER  IX. 

NUTATION  IN  LONGITUDE. 
70.  Longitude  affected  by  precession  of  the  equinoxes. — 71.   Precession  explained. — 72. 
Caused,  in  part,  by  the  moon's  attraction.— 73.  When  most  rapid.— 74.  Lunar  nutation, 
how  caused.— 75.  Equation  for  lunar  nutation  explained.— 76.  The  same   applied.— 77. 
Solar  nutation. 

CHAPTER  X. 

TRUE  TIME  LONGITUDES  AND  ANOMALIES. 
78.  Correction  in  the  time  of  new  moon. — 79.  Greenwich  time. — 80.  Correction  of  the  lon- 
gitudes.—81.  Synopsis.— 82.   Preliminary  equations  explained.— 83.  Application  of  the 
same.— 84.  Synopsis. 

CHAPTER  XL 

ELEMENTS  OF  AN  ECLIPSE. 
85.  List  of  elements.— 86.  Time  and  longitudes.— 87.  Obliquity  of  ecliptic  to  equator.— 88. 
Moon's  latitude.— 89.  Hourly  motions  of  sun  and  moon.— 90.  Sun's  apparent  semidiame- 
ter.— 91.  Moon's  semidiameter  and  parallax.— 92.  Semidiameter  of  earth's  shadow.— 93. 
Angle  of  moon's  path. — 94.  Sun's  declination. — 95.  Elements  collected. 

CHAPTER  XII. 

DELINEATION  OF  A  SOLAR  ECLIPSE. 
96.  Appearance  of  the  earth  as  viewed  from  the  sun,  while  the  latter  advances  from  the 
vernal  equinox  to  the  summer  solstice. — 97.  The  same,  while  it  advances  to  the  autumnal 
equinox. — 98.  The  same,  while  it  advances  through  the  winter  solstice  to  the  vernal  equi- 
nox.— 99.  Subject  illustrated  by  a  terrestrial  globe. — 100.  Relative  size  of  objects,  how 
measured  in  astronomy. — 101.  To  draw  the  northern  half  of  the  earth's  disc,  as  seen  from 


CONTENTS.  VII 

the  sun.  102.  To  find  the  position  of  the  north  pole  on  the  disc. — 103.  To  find  the  position 
of  any  given  place  at  noon  on  the  disc. — 104.  To  find  the  same  at  midnight. — 105.  To  find 
the  same  at  six  o'clock,  morning  and  evening. — 106.  To  find  the  same  at  any  other  hour, 
and  to  draw  the  apparent  diurnal  path  of  the  place. — 107.  The  construction  shows  the 
time  of  sunrise  or  sunset. — 108.  To  find  the  position  of  the  moon  when  new,  as  projected 
upon  the  earth's  disc. — 109.  To  draw  its  path  across  the  disc. — 110.  To  find  its  position  at 
any  time. — 111.  To  find  how  many  digits  will  be  eclipsed. — 112.  To  find  when  the  eclipse 
will  begin  and  end. — 113.  To  find  the  above  by  calculation. 

CHAPTER  XIII. 

CENTRAL  TRACK  OF  A  SOLAR  ECLIPSE. 
Hi.  Track  may  be  found  by  the  drawing  in  the  last  chapter. — 115.  A  better  method  by  a 
terrestrial  globe. — 116.  How  to  find  where  the  centre  of  the  moon's  shadow  first  strikes 
the  earth. — 117.  How  to  find  where  it  leaves  it. — 118.  How  to  find  where  the  eclipse  is 
central  at  any  given  time. — 119.  Track  may  be  found  more  accurately  still  by  calcula- 
tion.— 120.  How  to  find,  by  calculation,  where  the  centre  of  the  moon's  shadow  strikes  or 
leaves  the  earth. — 121.  How  to  find,  by  calculation,  where  the  eclipse  will  be  central  at 
any  given  time. — 132.  The  same,  more  fully  explained. — 123.  Results  of  the  calculation 
in  the  solar  eclipse  of  May,  1854.— 124.  General  description  of  the  track  of  that  eclipse. 

CHAPTER  XIV. 

DELINEATION  OF  A  LUNAR  ECLIPSE. 
125.  To  draw  the  earth's  shadow.— 126.  To  find  the  position  of  the  moon  when  full,  in  res- 
pect to  the  shadow.— 127.  To  draw  its  path.— 128.  To  find  the  position  of  the  moon  at  any 
lime  during  the  eclipse.— 129.  The  same  obtained  by  calculation.— 130.  Results  of  the 
calculation  in  the  lunar  eclipse  of  November,  1844. 


10 

earth.  The  angle  MES,  is  not  difficult  of  cpmputation,  being 
equal  to  SEA+MEC+AEC,  the  latter  of  which  is  equal  (Euc. 
1,  32)  to  ECB— EAB.  Hence,  MES=SEA+MEC+ECB— EAB, 
all  of  which  are  easily  found.  The  two  former  are  the  apparent 
semidiameters  of  the  sun  and  moon,  as  viewed  from  the  earth,  and 
the  two  latter  the  semidiameter  of  the  earth,  as  viewed  from  the 
moon  and  sun,  or,  respectively,  the  moon's  and  sun's  horizontal 
parallax.  On  account  of  the  distance  of  the  sun  and  moon  from 
the  earth  not  being  constant,  the  angle  MES  is  subject  to  a  varia- 
tion in  size,  being  sometimes  1°  38',  and  at  other  times  not  more 
than  1°  14'.  The  reader  will  perceive  that,  when  the  sun  and 
moon  are  in  conjunction,  the  angle  MES  is  the  moon's  latitude  ; 
and  the  conclusion  to  which  we  have  just  arrived,  may  be  express- 
ed thus  :  If  the  latitude  of  the  moon,  when  new,  is  less  than  1°  38' 
there  may  be  an  eclipse  of  the  sun,  and  if  it  is  less  than  1°  14'  there 
must  be  one. 

Fig.  2. 


4.  Again,  let  S  (Fig.  2)  represent  the  centre  of  the  sun,  E  that 
of  the  earth,  and  M  that  of  the  moon,  just  impinging  upon  the 
earth's  shadow.  The  angle  MET,  as  represented  in  the  figure,  is 
plainly  the  least  possible,  without  producing  a  partial  or  total 
eclipse  of  the  moon.  The  angle  MET=MEF+FET,  both  of 
which  can  be  easily  computed.  The  former  is  the  moon's  appa- 
rent simidiameter,  as  seen  from  the  earth,  and  FET  is  that  of  the 
section  of  the  earth's  shadow  that  eclipses  the  moon.  Now,  (Euc. 
1,  32,)  FET=BFE— FHE  and  FHE=AES— BAE ;  therefore, 
FET=BFE+BAE — AES,  the  two  former  of  which  are  the  lunar 
and  solar  parallaxes,  and  the  latter  is  the  sun's  apparent  semidiam- 
eter, as  seen  from  the  earth.  That  is,  the  apparent  semidiameter  of 
the  section  of  the  earth's  shadow  thai  eclipses  the  moon,  is  equal  to 
the  sum  of  the  parallaxes  of  the  sun  and  moon,  diminished  by  the 
sun's  apparent  semidiameter.  And  if  this  angle  be  increased  by 
the  moon's  apparent  semidiameter,  MEF,  we  shall  have  the  whole 
angle  MET,  which  is  the  least  latitude  the  moon  can  have  in  oppo- 


11 

sition,  without  being  eclipsed.  This  angle  varies  in  size,  like  the 
analagous  one  in  solar  eclipses,  just  described,  and  for  the  same 
reason.     Its  maximum  value  is  1°  4',  and  its  minimum  50'. 

5.  The  centre  of  ihe  section  of  the  earth's  shadow  that  eclipses 
the  moon,  must  be  situated  in  the  plane  of  the  ecliptic,  directly  op- 
posite to  the  sun,  and  must,  therefore,  be  at  the  same  distance  from 
one  of  the  moon's  nodes  that  the  sun  is  from  the  other.  Now,  we 
wish  to  know  how  near  the  sun,  in  its  annual  course,  may  approach 
to  one  of  the  moon's  nodes,  without  occasioning  eclipses  ;  or.  in 
other  words,  at  what  distance  from  the  node  the  moon's  track  and 
the  ecliptic  will  have  diverged,  so  as  to  be  from  1°  14'  to  1°  38' 
apart,  if  our  inquiry  relates  to  solar  eclipses — or,  from  50'  to  1°  4'» 
if  it  relates  to  lunar. 

6.  Let  AN  (Fig.  3)  represent  a 
portion  of  the  ecliptic,  BN  a  por- 
tion of  the  moon's  orbit,  SM  a  por- 
tion of  a  secondary  to  the  ecliptic, 
and  N  one  of  the  moon's  nodes.  Then,  in  the  right  angled  spheri- 
cal triangle,  SMN,  we  have  the  angle  SNM=5°  7'  47".9,*  and  for 
solar  eclipses,  the  arc  SM=1°  14'  to  1°  38'.  With  these  data,  we 
find  NS  to  be  from  13°  14'  to  19°  42',  according  to  the  value  we 
give  to  SM.  Hence,  if  the  sun  is  within  19°  42'  of  the  moon's 
node,  on  either  side,  at  the  time  of  new  moon,  it  may  be  eclipsed  ; 
and  if  it  is  within  13°  24',  it  must  be.  These  distances  are  called 
the  solar  ecliptic  limits. 

Since  it  takes  the  sun  more  than  a  lunar  month,  usually,  to  pass 
over  one  of  these  arcs,  it  follows,  that  it  must  be  eclipsed  at  every 
passage,  and,  consequently,  twice  a  year,  at  least.  It  may  be 
eclipsed  twice  during  one  passage ;  once,  just  after  it  enters  the 
ecliptic  limits,  and  again,  just  before  it  leaves  them  :  but,  if  so,  both 
of  the  eclipses  will  be  small,  and  not  central  upon  any  part  of  the 
earth. 

7.  For  lunar  eclipses,  the  arc  SM  is  50'  to  1°  4',  and,  by  the 
same  process  as  above,  we  find  the  lunar  ecliptic  limits  to  be,  irom 
7°  47"  to  13°  21'  on  each  side  of  the  node.f     They  embrace  an 

*  This  angle  is  subject  to  a  slight  variation,  amounting,  at  it3  maximum,  to  6'  47".lo. 
t  Baiiy. 


12 

arc  considerably  less  than  is  passed  over  by  the  sun  in  one  luna- 
tion, so  that  it  often  happens,  that  the  sun  passes  the  node  without 
there  being  any  lunar  eclipse. 

The  lunar  ecliptic  limits  being  so  much  less  than  the  solar, 
eclipses  of  the  moon  must  be  proportionably  less  frequent;  yet, 
since  a  lunar  eclipse  is  always  visible  over  half  the  earth's  surface, 
while  one  of  the  sun  can  be  seen  only  over  a  very  much  smaller 
section,  there  will,  on  an  average,  be  a  greater  number  of  visible 
eclipses  of  the  moon,  at  any  given  place,  than  of  the  sun, 

.8.  As  the  moon's  nodes  are  180°  apart,  or,  in  opposite  points  of 
the  ecliptic,  the  interval  between  eclipses  occurring  at  one  node, 
and  those  occurring  at  the  other,  must  be  about  six  months.  Also, 
since  the  nodes  move  backward  about  19°  in  each  year,  eclipses 
must  happen,  on  an  average,  nearly  three  weeks  earlier  every  year 
than  they  did  on  the  year  preceding.  The  reader  will  see  that 
these  conclusions  are  verified  by  past  experience,  if  he  will  take  the 
trouble  to  examine  the  almanacs  of  former  years. 


CHAPTER  II. 

MEAN    TIME    OF  AN    ECLIPSE,  AND    THE    MEAN    LONGITUDES    AND    ANOMA- 
LIES   OF    THE    SUN    AND    MOON. 

9.  The  sun,  in  its  apparent  annual  course,  leaves  the  vernal  equi- 
nox where  its  longitude  is  0°,  about  the  21st  of  March,  and  moves 
eastward,  towards  the  moon's  nodes,  about  1°  each  day.  Conse- 
quently, it  must  arrive  at  either  node,  in  about  as  many  days  after 
the  21st  of  March,  in  any  given  year,  as  the  longitude  of  the  node, 
in  that  year,  contains  degrees.  At  the  next  new  moon  before  or 
after  the  date  thus  found,  (more  frequently  the  former.)  there  will 
be  an  eclipse  of  the  sun.  It  is  probable,  though  not  certain,  that 
there  will  also  be  a  lunar  eclipse  at  the  nearest  full  moon.  It  will 
occur  then  if  at  all  at  that  passage  of  the  node. 

10.  The  velocity  of  the  motions  of  sun  and  moon  in  their  respect- 
ive orbits,  is  variable ;  but  it  is  more  convenient  in  astronomical 
calculations  to  regard  it  as  uniform,  and  to  make  the  necessary  cor- 
rections for  the  inequalities  afterward. 


13 

11.  To  explain  the  method  of  calculating  the  time  of  an  eclipse, 
we  will  take,  as  an  example,  the  solar  eclipse  that  will  occur  when 
the  sun  passes  the  moon's  ascending  node,  in  the  year  1854,  Ta- 
ble 2d,  at  the  end  of  this  volume,  contains  the  time  of  new  moon 
in  March  of  that  year,  as  well  as  of  every  other  during  the  present 
century,  and  the  longitude  of  the  sun,  moon,  and  moon's  ascending 
node,  all  calculated  on  the  supposition  of  a  uniform  rate  of  motion. 
It  also  gives  the  .mean  anomolies  of  the  sun  and  moon,  i.  e.,  the  dis- 
tance of  each  from  its  perigee.  The  longitude  of  the  descending 
node  may  be  found  by  adding,  or  subtracting,  180°  to,  or  from, 
that  of  the  ascending  node.  We  might  compute  all  these  quanti- 
ties from  the  data  given  in  table  1st,  but  table  2d  supercedes  the 
necessity.  Although  some  of  them  will  not  be  used  in  this  chapter, 
it  is  most  convenient  to  take  them  all  out  together,  and  write  them 
as  below.  The  longitude  of  the  node  on  that  year  (see  right  hand 
column  of  the  table)  is  64°.2158  ;  consequently,  the  sun  will  arrive 
at  it  about  64  days  after  the  21st  of  March,  which  carries  the  time 
to  May  24th.  The  eclipse  in  question  will  occur  at  the  new  moon 
nearest  that  time.  Entering  table  3d,*'  we  next  take  out  such  a 
number  of  lunations  (in  this  ease,  two)  as,  when  added  to  the  time 
of  new  moon  in  March,  will  bring  it  near  to  the  time  when  the  sun 
reaches  the  moon's  node,  and  write  it  down,  with  the  longitudes 
and  anomalies,  under  the  corresponding  quantities  already  taken 
from  table  2d.  These  must  be  added  together,  (with  the  exception 
of  the  right  hand  column,  where  the  lower  number  must  be  sub- 
tracted from  the  upper,  because  the  motion  of  the  node  is  retro- 
grade,) and  we  thus  obtain  the  time  of  mean  new  moon  in  May. 
The  following  shows  the  operation  : — 


Mean  new  moon  in  March,... 

Add  two  lunations, 

Mean  new  moon  in  May, 


Time. 


h.  ?n.    s, 
28  10  42  50 
59     1  28    6 


26  12  10  5t 


Sun's 
Anom- 
aly. 


85.586 
58.21 1 


143.797 


Sun's 
Longi- 
tude. 


6°0194 
58.2135 


Moon's 
Anom- 
aly. 


93.665 
51.634 


64.2329|145.299 


Moon's  I  Longi- 
Longi-  tude  of 
tude.       Node. 


6.0194  64°2158 
58.2135l-3.1275 


64.2329  i  61.0883 


Table  5th  shows  the  month  and  day  to  which  any  number  of 
days  found  by  the  foregoing  addition  corresponds. 

At  this  stage  of  the  calculation,  it  is  well  to  compare  the  longi- 
tude of  the  sun  and  moon  with  that  of  the  node,  and  if  they  do  not 


*  Table  3d  shows  the  length  of  any  number  of  mean  lunations,  from  one  to  thirteen, 
with  the  mean  motions  of  the  sun  and  moon,  both  in  longitude  and  anomaly,  during  the 
same  ;  also,  the  mean  motion  of  the  moon's  nodes. 


14 

differ  more  than  20°,  there  may  be  an  eclipse,  though  there  will  not, 
probably,  be  one,  if  the  difference  is  over  I620.  If  the  difference  is 
too  great,  it  shows  that  too  many  lunations  were  added,  or  too  few, 
and  a  correction  must  be  made  accordingly.  A  difference  of  over 
11°  shows  that  another  eclipse,  at  that  passage  of  the  node,  is  pos- 
sible, but  not  probable,  unless  it  is  as  much  as  14°. 

12.  At  the  time  of  new  moon,  the  longitudes  of  the  sun  and  moon 
must  be  equal,  and,  according  to  our  calculations,  they  are  so  at 
the  time  of  the*mean  new  moon  in  May  just  found.  But  this  is  on 
the  supposition,  that  their  motions  were  uniform.  To  find  whether 
or  not  their  longitudes  are  truly  equal,  we  shall  proceed,  in  the  fol- 
lowing chapters,  to  compute  them,  taking  into  account  all  the  chief 
inequalities  in  their  motions ;  and,  if  they  come  out  alike,  the  time 
of  new  moon  is  correctly  found  ;  otherwise,  we  shall  have  to  add 
or  subtract  such  an  amount  of  time,  as,  with  the  relative  velocities 
of  the  sun  and  moon,  at  the  time,  will  render  them  equal.  As  a 
preparatory  step,  it  is  necessary  to  know,  more  accurately,  the 
the  moon's  anomaly,  and  the  longitude  of  its  node. 

The  progressive  motion  of  the  moon's  perigee,  and  the  retro- 
grade motion  of  its  nodes,  being  both  caused  by  the  sun's  attrac- 
tion, are  most  rapid  when  the  sun  is  in  its  perigee,  and  constantly 
grow  slower  and  slower,  till  the  sun  reaches  its  apogee,  where  the 
motion  becomes  the  slowest.     Consequently,  as  the  sun  leaves  its 
perigee,  the  moon's  perigee  immediately  gets  before,  and  its  nodes 
behind  their  mean  place,  and   continue  so  till   the  sun   reaches  its 
apogee,  when,  owing  to  the  diminished  rate  of  motion,  their  mean 
and  true  places  again  coincide.     The  contrary  takes  place  when 
the  sun  is  in  the  other  half  of  its  orbit.     Hence  it  is  apparent,  that 
the  moon's  anomaly,  being  reckoned  from  its  perigee,  must  be  less 
than  the  mean  when  the  sun's  anomoly  is  less  than  180°,  and  great- 
er when  greater  ;  showing  that  something  must  be  subtracted  from 
the  moon's  anomaly  in  the  former  case,  and  added  in  the  latter. 
The  same  must  also  be  true  of  the  longitude  of  the  moon's  nodes. 
These  facts  are  indicated  in  Tables  6th  and  7th,  by  the  signs  — 
and  +  placed  at  the  head  of  the  column  containing  the  argument. 
By  the  term  argument,  is  meant  that  quantity  on  which  others 
depend,  and  which  determines  their  value.     Thus,  in  this  case,  the 
sun's  anomaly  determines  what  correction  must  be  applied  to  the 
moon's  anomaly,  or  to  the  longitude  of  its  node,  and  is,  therefore, 
the  argument. 


15 

13.  Entering  tables  6th  and  7th,  with  the  sun's  anomaly  as  an 
argument,  we  will  take  out  the  corrections  which  the  foregoing 
considerations  show  to  be  necessary,  and  which  are  denominated 
Annual  Equations  of  the  moon's  perigee  and  node,  (taking  care  to 
make  a  proper  allowance  for  the  odd  degree  and  decimals  of  the 
anomaly,  as  the  tables  give  the  equation  only  for  every  two  de- 
grees,) and  apply  the  former  to  the  moon's  anomaly,  and  the  latter 
to  the  longitude  of  the  node,  according  to  the  sign  -f-  or  —  at  the 
head  of  the  column  of  the  argument.  Observe  in  these,  and  most 
of  the  other  tables,  that  the  unit  figure  of  the  argument  is  placed 
at  the  top  or  bottom  of  the  table,  and  the  other  figures  at  the  right 
or  left.  When  the  latter  is  found  at  the  left,  we  must  look  for  the 
former  at  the  top,  but  when  the  latter  is  at  the  right,  the  former  must 
be  sought  for  at  the  bottom.  Opposite  the  latter,  and  in  the  same 
column  with  the  former,  the  equation  is  found.  Thus,  in  our 
eclipse,  the  sun's  anomaly  being  143°. 797,  we  look  for  the  number 
14  in  the  left  hand  column,  and  for  3  at  the  top.  But,  since 
the  latter  number  is  not  found,  the  equation  being  given  in  the  ta- 
ble only  for  142°  and  144°,  we  must  note  the  difference  between 
these  equations,  and  take  a  proper  proportion  of  it  for  the  excess  of 
the  argument  over  142°,  viz.  1°.797.  By  this  process,  we  find  the 
annual  equation  of  the  perigee  to  be  — 0°.221  ;  and  of  the  node 
— 0°.875.  These,  applied  to  the  moon's  anomaly,  and  the  longitude 
of  the  node,  make  the  former  145°.078,  and  the  latter  61°.0008. 


CHAPTER  III. 

EQUATION  OP  THE  CENTRE. 


14.  The  first  inequality  in  the  apparent  motions  of  the  sun  and 
moon  that  claims  attention,  results  from  the  eliptical  form  of  their 
orbits,  in  consequence  of  which  their  motion  is  accelerated  while 
passing  from  apogee  to  perigee,  and  retarded  in  the  other  half  of 
their  orbits,  moving  quickest  in  perigee,  and  slowest  in  apogee. 
The  reason  of  this  it  is  not  difficult  to  discover. 


16 

Let  AGH,  &c.  (Fig.  4)  represent  the 
moon's  eliptical  orbit,  A  and  B  the  apsides, 
and  E  the  earth.  Let  the  moon  start  from 
perigee,  at  A,  with  its  swiftest  motion,  and 
consequently  with  its  greatest  centrifugal 
force.  The  attraction  of  the  earth,  at  E, 
not  being  able  to  retain  it  at  that  distance, 
G'  H  it  immediately  begins  to  recede  along  the 

curve  AGH.  Being  constantly  pulled  back  by  the  earth's  attrac- 
tion, since  the  angle  EGH  is  obtuse,  its  motion  is  retarded,  and 
when  it  approaches  the  apogee,  at  B,  its  velocity  has  become  so 
much  diminished,  that  the  attractive  force  of  the  earth  prevents  it 
from  receding  further.  It  thus  arrives  at  apogee  with  its  slowest 
motion.  Leaving  B  with  a  weak  centrifugal  force,  the  superior 
attractive  power  of  the  earth  at  E,  immediately  begins  to  draw 
the  moon  toward  itself  along  the  curve  BOD,  constantly  hurrying 
it  onward  at  every  point,  as  O,  the  angle  EOD,  contained  between 
the  direction  toward  which  it  is  drawn,  and  that  toward  which  it 
moves  being  now  acute.  By  the  time  it  reaches  A,  its  velocity 
becomes  so  much  increased,  that  it  is  prepared  again  to  leave  peri- 
gee in  the  same  condition  as  at  first,  to  pursue  another  similar 
round. 

15.  We  shall  arrive  at  the  same  conclusion,  if  we  apply  the  prin- 
ciple, that  when  a  body  is  retained  in  its  orbit  by  a  force  directed 
toward  a  fixed  point,  as  the  moon  is  toward  the  earth  in  this  case, 
the  radius  vector  must  describe  equal  areas  in  equal  times.  Hence 
the  moon  must  move  slower  when  it  is  near  B,  than  when  it  is 
near  A,  in  about  the  same  ratio  that  its  distance  from  the  earth  is 
greater.  Not  only  is  the  absolute  velocity  greatest  in  perigee,  but 
the  angular  velocity,  with  which  only  we  are  now  concerned,  is 
rendered  still  greater,  by  reason  of  the  diminished  distance. 

The  same  reasoning  will  apply,  in  every  respect,  to  the  earth 
revolving  in  an  eliptical  orbit  round  the  sun,  and  hence  to  the  ap- 
parent motion  of  the  sun  round  the  earth. 

16.  If  we  consider  the  mean  place  of  the  sun  and  moon  to  be 
their  true  one,  when  in  perigee,  it  will  be,  also,  when  in  apogee ; 
because  each  half  of  the  orbit  is  described  in  the  same  time ;  but, 
as  they  start  from  perigee  with  their  swiftest  motion,  they  thus  get 
ahead  of  their  mean  place,  nor  do  they,  though  constantly  retarded, 


17 

lose  what  they  had  gained,  till  the  moment  they  arrive  at  apogee. 
Also,  as  they  pass  the  apogee  with  their  slowest  motion,  they  di- 
rectly get  behind  their  mean  place,  and  it  is  not  till  the  moment 
they  reach  the  perigee,  that  the  continued  acceleration  of  their 
motion,  in  this  half  of  their  orbits,  enables  them  to  gain  up  what 
they  had  thus  lost.  Consequently,  they  must  always  be  ahead  of 
their  mean  place,  when  in  the  former  part  of  their  orbits,  and  behind . 
it  in  the  latter.  This  difference  between  the  mean  and  true  place 
of  the  sun  or  moon,  is  termed  the  equation  of  its  centre. 

17.  It  is  necessary  to  know  in  which  half  of  their  respective 
orbits  the  sun  and  moon  are  found  at  the  time  of  our  predicted 
eclipse  :  for,  if  either  is  moving  from  perigee  to  apogee,  i.  e.,  if  its 
anomaly  (4)  is  less  than  180°,  we  must  add  something  to  its  longi- 
tude already  found,  (11,)  but  subtract  if  in  the  other  half  of  its  orbit, 
i.e.,  if  its  anomaly  is  over  180°.  The  manner  of  computing  the 
precise  amount,  will  occupy  our  attention  in  another  chapter.  It 
is  sufficient,  here,  to  remark,  that  if,  at  any  time,  a  line,  OC,  be 
drawn  from  the  mean  place  of  the  moon  to  the  centre  of  the  elipse, 
the  angle  EOC,  which  this  line  makes  with  the  radius  vector,  is 
very  nearly  equal  to  one-half  the  angular  distance  by  which  the 
moon  is  before  or  behind  its  mean  place  :  so  nearly  that  some  au- 
thors have  given  this  as  a  method  of  computing  the  equation  of  the 
centre.  We  shall  use  it  hereafter  as  a  convenient  approxima- 
tion. For  the  present,  we  will  dispense  with  the  labour  of  compu- 
tation, and  take  the  equation  directly  from  tables  already  prepared. 

By  the  calculations  in  the  last  chapter,  (11  and  13,)  we  found 
that  the  sun's  anomaly  was  143°.797,  and  the  moon's,  as  corrected, 
145°.078.  It  appears,  then,  that  each  is  moving  from  perigee  to 
apogee,  and  is,  therefore,  ahead  of  its  mean  place,  so  that  we  must 
add  something  to  their  respective  longitudes  ;  with  these  anoma- 
lies, respectively,  as  arguments,  we  may  now  enter  tables  8th  and 
9th,  in  the  same  manner  as  we  did  tables  6th  and  7th,  and  take  out 
the  equation  of  the  centre  of  the  sun,  and  of  the  moon,  applying  the 
former  to  the  the  sun's  longitude,  and  the  latter  to  the  moon's.  The 
equations  we  find  to  be  +  1M165  and  +3°.4154,  and  the  resulting 
longitudes  65°.3494  and  67°.6483. 

2 


18 


CHAPTER  IV. 

PERTURBATIONS  IN  THE  MOON's  MOTION. ANNUAL  AND  SECULAR  EQUA- 
TIONS OF  LONGITUDE. 

18.  In  the  preceding  calculations,  we  first  regarded  the  sun  and 
.moon  as  revolving  in  circular  orbits,  with  uniform  angular  velocity  ; 
then,  in  elipses,  describing  equal  areas  by  the  radius  vector,  in 
equal  times  :  but  neither  of  these  suppositions  is  strictly  true.  The 
sun's  attraction  disturbs  the  motion  of  the  moon  round  the  earth, 
producing  numerous  inequalities. 

The  method  of  calculating  these,  will  be  treated  of  in  another 
place.  It  will  be  sufficient  for  our  purpose,  here,  simply  to  give 
the  theory  of  them,  and  then  take  the  corresponding  corrections 
from  the  tables. 

19.  The  most  obvious  effect  of  the  sun's  attraction,  is  to  draw 
the  moon  away  from  the  earth,  and  thus  enlarge  its  orbit.  If  this 
influence  were  always  the  same,  it  would  occasion  no  inequality : 
but  when  the  sun  is  in  perigee,  it  is  nearer  to  the  earth,  and  conse- 
quently to  the  moon,  than  at  other  times  ;  and  the  moon,  therefore, 
will  be  more  attracted  by  it.  The  moon  being  thus  drawn  farther 
away  from  the  earth,  when  the  sun  is  in  this  situation,  and  its  peri- 
odic time  consequently  increased,  it  must  fall  behind  its  mean  place. 
And  although  the  attractive  force  of  the  sun  diminishes  as  it  leaves 
perigee,  allowing  the  moon  to  contract  its  orbit  and  lessen  its  peri- 
odic time,  it  will  not  gain  up  what  it  had  lost  till  the  moment  the 
sun  reaches  apogee.  By  similar  reasoning,  we  may  see  that  the 
moon  must  always  be  in  advance  of  its  mean  place,  so  far  as  this 
cause  is  concerned,  when  the  sun  is  in  the  other  half  of  its  orbit. 
There  must  then  be  applied  to  the  moon's  longitude  a  correction 
depending  on  the  sun's  anomaly  ;  .being  additive  when  the  anomaly 
is  less  than  180°,  and  subtractive  when  it  is  more.  Such  a  correc- 
tion is  supplied  in  table  10th,  entering  which,  with  the  degrees  of 
the  sun's  mean  anomaly,  in  the  manner  described  in  article  13th, 
we  find  the  equation  to  be  -.1118,  which  is  to  be  applied  both  to 
the  moon's  longitude  and  anomaly,*  for  the  cause  we  have  been 
considering  obviously  affects  both  alike.     The  same  is  true  of  most 

*  The  anomaly  contains  but  three  decimal  places ;  hence,  in  applying  the  corrections 
to  it,  the  3d  figure  is  given  according  to  its  nearest  value. 


d9 

of  the  other  corrections  that  remain  to  be  applied.  The  longitude, 
as  already  obtained,  (17,)  is  67°.6483,  and  the  anomaly  (13) 
145°.078.  Applying  trie  above  equation,  they  become  67°.5365, 
and  144°.9G6  respectively. 

20.  Connected  with  the  foregoing  inequality  in  the  moon's  mo- 
tion, there  is  another  of  great  historical  interest,  from  the  theories 
to  which  it  formerly  gave  rise,  viz.,  the  acceleration  of  the  moon's 
mean  motion.  It  is  too  small  to  be  discovered  by  direct  observa- 
tion, but  becomes  quite  sensible  in  the  lapse  of  ages. 

21.  Dr.  Haliey,  wishing  to  know  the  precise  length  of  a  lunation, 
went  back  to  the  ancient  Chaldean  observations,  intending  to  ascer- 
tain how  many  new  moons  had  occurred  between  that  time  and 
his  own,  and  then  to  divide  the  time  by  this  number,  which  would 
give  the  average  length  of  each.  But  he  was  surprised  to  find 
that  a  lunation  in  those  days  was  considerably  longer  than  now. 
By  comparing  the  Chaldean,  Alexandrian,  Arabian,  and  the  pre- 
sent observations,  he  found  that  the  lunar  period  grew  successively 
shorter. 

22.  Astronomers  doubted  the  fact  when  it  was  first  announced  ; 
but  when  they  became  satisfied  of  its  truth,  they  set  themselves  to 
work  to  account  for  it.  The  most  probable  theory  was,  that  the 
moon  revolved  in  a  resisting  medium,  which  would  cause  it  gradu- 
ally to  fall  toward  the  earth,  and  thus,  by  reducing  the  size  of  the 
orbit,  make  the  periodic  time  less.  It  must  seem  paradoxical  to 
those  who  have  not  thought  upon  the  subject,  that  such  a  cause 
could  produce  the  effect  in  question ;  and  that  the  retarding  of  the 
motion  could  make  it  revolve  in  less  time.  But  it  should  be  con- 
sidered, that  by  diminishing  the  moon's  velocity,  its  centrifugal 
force  is  diminished  in  a  more  rapid  ratio,  which  would  allow  the 
earth  to  draw  it  nearer  to  itself,  and  reduce  the  size  of  the  orbit. 
And  it  is  demonstrable,  that  the  gain  in  time  from  the  latter  cir- 
cumstance, would  more  than  counterbalance  the  loss  from  the  for- 
mer ;  so  that  on  the  whole,  the  moon's  period  would  be  shortened. 

The  objection  to  this  theory  is,  that  comets,  which  are  proved 
to  be  extremely  light  bodies,  pass  through  this  medium  w^ith  little 
or  no  resistance.     Hence  it  was  inferred  that  the  cause,  if  it  exist- 


20 

ed  at  all,  was  not  sufficient  to  produce  the  effect  of  which  we  arc 
speaking. 

Other  theories  were  advanced,  but  none  were  satisfactory  ;  and 
it  was  reserved  for  La  Place  to  explain  the  true  reason  of  this  ac- 
celeration of  the  moon's  period,  about  sixty  years  ago. 

23.  Owing  to  the  attraction  of  the  other  planets,  the  earth's  orbit 
is  gradually  becoming  less  and  less  eliptical,  or,  nearer  and  nearer 
to  a  circle  ;  so  that  the  sun  is  every  year  about  39|  miles  nearer 
the  centre  of  the  elipse  than  it  was  on  the  year  before.  At  this 
rate,  the  earth's  orbit  would  become  a  circle  in  40,315  years  ;  an 
event,  however,  that  can  never  take  place,  for  long  before  such  a 
period  shall  elapse,  the  change  of  which  we  are  speaking,  and 
which  is  only  an  inequality  of  long  period,  will  have  reached  its 
limit,  when  the  eccentricity  of  the  orbit  would  again  increase. 

24.  If  it  can  be  shown  that  the  sun's  attraction  diminishes  the 
moon's  gravity  toward  the  earth,  and  thus  increases  the  periodic 
time,  more  than  it  would  do  if  the  earth  revolved  in  a  circle  at  the 
same  mean  distance,  it  is  manifest  that  so  long  as  the  change  in  the 
shape  of  the  earth's  orbit,  of  which  we  have  just  spoken,  goes  on, 
the  moon's  periodic  time  must  grow  less  and  less. 

Let  ADBE  (Fig.  5)  represent  an  elipti- 
cal orbit,  S  the  attracting  body,  placed  in 
one  of  the  foci,  C  the  centre  of  the  elipse, 
and  F  the  other  focus.  It  can  be  demon- 
strated, that  the  mean  distance  of  S  from  all 
points  in  the  orbit,  is  equal  to  AC  or  CB. 
Take  any  two  points  in  the  orbit  G  and  H, 
equidistant  from  B  and  A.  We  propose  to  prove  that  the  average 
attraction  of  S  upon  the  moving  body,  when  at  these  points,  is 
greater  than  it  is  when  the  body  is  at  its  mean  distance.  And 
since  these  are  any  points  in  the  arcs  AD  and  DB,  if  we  prove  it 
for  them,  we  prove  it  for  the  whole  orbit. 

Join  GS,  GF  and  HS,  and  let  SG  bear  any  ratio,  other  than  that 
of  equality,  to  GF;  say  6  :  4.  Then,  since  by  the  properties  of  the 
elipse  SG-f  GF=AB=2CB,  it  follows  that  the  ratio  of  SG  to  CB 
is  6 :  5,  and  of  GF,  or  its  equal  HS,  to  CB,  4  :  5.  Therefore,  since 
the  force  of  gravity  is  inversely  proportioned  to  the  square  of  the 
distance,  the  attraction  at  the  former  point  will  be  §£,  and  at  the 


21 

latter  f- f  of  what  it  is  at  the  mean  distance.     The  average  between 
them  is  fff  of  the  attraction  at  the  mean  distance, — exceeding  it 

Now  the  earth's  orbit  is  much  nearer  circular  than  we  have 
supposed  this  to  be,  and  the  excess  of  attraction  must  be  propor- 
tionably  less :  but  still  there  must  be  an  excess,  so  long  as  it  is 
eliptical  at  all.  Hence,  as  the  earth's  orbit  becomes  nearer  circular, 
the  sun's  attraction  upon  it,  and  consequently  upon  the  moon,  must 
continually  grow  less,  allowing  the  orbit  of  the  latter  to  contract. 
This  would  diminish  the  periodic  time,  and  produce  the  very  effect 
that  excited  so  much  wonder  in  the  mind  of  Dr.  Halley,  and  the 
astronomers  of  his  time. 

25.  The  tables  for  this  work  are  based  on  the  moon's  motion,  as 
it  existed  in  the  year  1800,  and  we  must,  therefore,  add  to  its  lon- 
gitude and  anomaly  the  amount  gained  since  that  time,  from  the 
cause  just  explained.  Table  11th  contains  the  required  correction, 
calculated  at  intervals  of  five  years,  during  the  present  century. 
Look  for  the  year  in  the  left  hand  column  of  the  table,  except  the 
unit  figure,  which  is  placed  at  the  top,  and  opposite  to  the  former, 
and  under  the  latter  will  be  found  the  correction  required,  express- 
ed in  decimals  of  a  degree,  the  first  two  places,  which  are  ciphers, 
being  omitted. 

The  correction  for  the  year  1854  is  .0009,  which,  added  to  the 
longitude  and  anomaly  already  found,  (19,)  makes  the  former 
67°.5374,  and  the  latter  144°.967. 


CHAPTER  V. 

PERTURBATIONS  IN  THE  MOOn's    MOTION,  CONTINUED. VARIATION. 

26.  The  moon's  motions  grow  more  complicated  the  farther  we 
proceed.  To  investigate  them  thoroughly,  is  nothing  less  than  a 
solution  of  the  famed  Problem  of  the  Three  Bodies.  The  moon's 
orbit,  which  we  first  regarded  as  a  circle,  and  then  an  elipse,  we 
shall  now  find  to  be  neither  a  circle  nor  an  elipse,*  but  an  irregular 

*  This  statement  seems  to  conflict  with  former  ones,  where  the  eliptical  form  of  the 
moon's  orbit  was  asserted;  but  its  mean  shape  was  then  intended,  without  taking  into 
account  the  irregularities. 


22 

oval  shaped  figure,  which  is  constantly  changing  its  form.  The 
prospect  before  us,  in  trying  to  reduce  such  irregularities  to  order, 
so  as  to  see  their  precise  influence  on  the  moon's  longitude,  is  suffi- 
ciently appalling,  but  nevertheless,  let  us  not  be  deterred  from  the 
attempt. 

To  avoid  misapprehension,  it  ought  perhaps  here  to  be  remark- 
ed, that  these  irregularities  are  not  of  such  a  nature  as  to  set  aside 
our  previous  work,  but  only  show  that,  under  some  circumstances, 
they  may  occasion  necessary  corrections. 

•27.  And  first  let  us,  in  this  chapter,  see  what  the  shape  of  the 
orbit  would  be,  and  how  the  moon  would  revolve  in  it,  on  the  sup- 
position that  it  was  originally  a  circle  round  the  earth,  but  drawn 
out  of  shape  by  the  sun's  attraction.  We  shall  in  this  way  disco- 
ver the  cause  of  an  observed  inequality  in  the  moon's  motion,  de- 
nominated variation,  and  discovered  by  Tycho  Brahe,  A.  D.  1590. 
The  modifications  that  the  orbit  would  undergo,  by  supposing  the 
original  figure  an  elipse  instead  of  a  circle,  will  occupy  our  atten- 
tion in  the  next  chapter. 

28.  Let  S  (Fig.  6)  represent  the  sun,  E  the  earth,  and  ADCB 
the  moon's  orbit ;  and  let  us  suppose,  for  a  moment,  that  the  moon, 
retains  a  circular  orbit.  Let  D  represent  the  place  of  the  moon  at 
conjunction,  C  at  apposition,  and  A  and  B  when  it  is  at  the  same 
distance  from  the  sun  that  the  earth  is,  or  very  nearly  in  quadra- 
ture. 

First,  let  the  moon  be  at  A  or  B,  in  which  case  the  moon  and 
earth  being  equally  distant  from  the  sun,  must  be  equally  attracted 
by  it,  and  consequently  there  would  be  no  tendency  to  change 
their  direction  from  each  other,  but  only  to  draw  them  nearer  to- 
gether, which  would  be  precisely  equivalent  to  increasing  the  earth's 
power  of  gravity. 

Next  let  it  be  in  conjunction  at  D.  Now  the  earth  and  moon 
are  both  in  the  same  direction  from  the  sun  ;  but  the  moon  being 
nearest  is  more  attracted,  in  the  inverse  ratio  of  the  square  of  the 
distance,  i.  e.,  SE2  :  SD2-  The  only  effect,  therefore,  is  to  draw 
the  moon  directly  away  from  the  earth,  by  virtue  of  the  difference 
in  the  attractive  forces,  which  would  be  equivalent  to  diminishing 
the  earth's  attraction. 


29.  Again, — suppose  the  moon  at  any  point  M  in  the  quadrant 
AD.  Being  nearer  to  the  sun  than  the  earth  is,  it  is  more  attract- 
ed by  it,  and  the  effect  is  nearly*  the  same  as  though  the  earth  was 
not  attracted  at  all,  but  the  moon  drawn  along  the  line  MS  by  a 
force  equal  to  the  difference  of  the  attractions.  The  direction  of 
this  force,  making  an  acute  angle  with  that  in  which  the  moon 
moves,  must  accellerate  the  motion  in  its  orbit;  and  the  same 
would  be  true  of  every  point  in  the  quadrant  AD.  If  the  moon 
were  at  M'  any  point  in  the  quadrant  DB,  the  difference  of  attrac- 
tions acting  along  the  line  M'S,  would  tend  to  retard  its  motion. 

Once  more :  let  the  moon  be  at  any  point  M"  in  the  quadrant 
BC.  Being  further  from  the  sun  than  the  earth  is,  it  is  less  attract- 
ed by  it,  which  is  nearly  as  though  it  were  drawn  in  the  opposite 
direction,  along  the  line  M"L.  The  effect  would  be  to  accelerate 
the  motion,  in  nearly  the  same  manner  as  in  the  quadrant  AD.  If 
the  moon  was  at  any  point  M'"  in  the  quadrant  CA,  the  difference 
of  attractions  acting,  as  it  were,  along  the  line  M'"N,  would  retard 
its  motion. 

Thus  the  moon  is  alternately  accelerated  and  retarded  in  the 
differant  quadrants ;  moving  swiftest  in  syzygy  and  slowest  in 
quadrature.  Hence,  from  this  cause  alone,  the  moon  would  be  in 
advance  of  its  mean  place  while  passing  from  syzygy  to  quadrature, 
and  behind  it  while  passing  from  quadrature  to  syzygy. 

80.  But  the  above  is  not  the  only  reason.  We  have  thus  far,  in 
the  present  investigation,  supposed  the  moon's  orbit  to  retain  its 
circular  form,  notwithstanding  the  disturbing  influence  of  the  sun  : 
but  this  is  not  possible.  To  retain  a  body  in  a  circular  orbit,  the 
centripetal  and  centrifugal  forces  must  be  equal.  But  we  have 
just  seen  that  the  velocity  in  syzygy,  as  at  C  and  D,  (Fig.  7,)  is 

*  Sufficiently  near  for  our  present  purpose.  The  subject  will  be  investigated  more 
critically  hereafter. 


24 


greater  than  at  A  and  B :  and  as  the  centrifugal  force  is  propor- 
tioned to  the  square  of  the  velocity,  it  must  be  greater.  On  the  other 
hand,  it  was  shown  (28)  that  the  sun's  attraction  diminished  the 
moon's  gravity  toward  the  earth  in  syzygy,  as  at  C  and  D,  and  in- 
creased it  in  quadrature,  as  at  A  and  B.  Taking  both  these  facts 
into  consideration,  it  is  man.  Fig.  7. 

ifest  that  at  A  and  B,  the 
centripetal  force  must  con- 
siderably exceed  the  centri- 
fugal, while  at  C  and  D,  the 
centrifugal  will  be  the  great- 
est, which  would  cause  the 
moon's  track  to  fall  within  the  circle  at  the  former  points,  as  to  L 
and  N,  and  without  it  in  the  latter,  as  to  F  and  G. 

The  effect  would  be  to  Fig.  8. 

throw  the  orbit  into  some- 
thing such  a  shape  as  is 
represented  in  Fig.  8,  viz: 
a  kind  of  oval,  with  its 
longest  diameter,  AB,  at 
right  angles  to  line  ES, 
drawn  from  the  earth  to 
the  sun. 

Will  this  alteration  in  the  the  shape  of  the  moon's  orbit  affect  its 
longitude  ?  To  aid  us  in  this  investigation,  we  will  circumscribe 
the  oval  by  a  circle ;  and  to  make  the  illustration  more  striking, 
we  will  suppose  the  oval  very  much  flattened,  so  as  nearly  to  coin- 
cide with  AB,  as  in  Fig.  9.  Now,  if  the  arc  AF  be  divided  in  any 
given  ratio  at  the  point  L,  and  LE  be  Fi£-  9- 

drawn,  it  will  cut  the  arc  AD  by  no  means 
in  the  same  ratio.  AM  will  bear  a  much 
greater  ratio  to  MD  than  AL  to  LF. 
Hence,  if  two  bodies,  whose  periodic 
times  were  equal,  should  start  from  A  atf| 
the  same  time,  and  move  with  uniform 
velocity,  one  in  the  circle  ALF,  and  the 
other  in  the  oval  AMD,  the  former  would 
arrive  at  L  before  the  latter  would  at  M, 
leaving  it  behind  perhaps  at  N.  The  same  reasoning  may  be  ap- 
plied to  the  other  quadrants,  though  with  the  opposite  effect  in  DB 


25 

and  CA,  when  it  will  show  that  the  moon  must  be  in  advance  of 
its  mean  place. 


31.  There  are  two  reasons,  then,  why  the  moon  will  be  behind 
its  mean  place  when  passing  from  quadrature  to  syzygy,  but  in 
advance  of  it  while  passing  from  syzygy  to  quadrature  ;  1st,  from 
its  unequal  motion,  (29,)  and  2d,  from  the  shape  of  its  orbit.  The 
maximum  effect  of  the  former  to  change  the  moon's  place,  is  from 
9'  17"  to  10'  15",  and  the  latter  from  23'  56"  to  20'  52",  according 
to  the  distance  of  the  earth  from  the  sun.  When  at  its  mean  dis- 
tance, the  maximum  effects  are  9'  46"  for  the  former,  and  25'  24" 
for  the  latter,  amounting  to  35'  10"  for  both  united.     * 


32.  If  the  four  quadrants  were  perfectly  symmetrical,  a  table 
showing  the  correction  required  for  each  degree  in  one  quadrant, 
would  answer  for  all  the  rest ;  only  the  equation  would  be  additive 
when  the  moon  is  passing  from  syzygy  to  quadrature,  i.  e.,  in 
the  arcs  DB  or  CA,  and  subtractive  when  it  is  passing  from  quad- 
rature to  syzygy,  i.  e.,  in  the  arcs  AD  and  BC.  But  there  is  a 
slight  difference ;  for,  1st,  the  disturbing  influence  is  a  little  less  in 
the  half  of  the  orbit  nearest  the  sun  than  in  the  other  half,  the  dif- 
ference of  the  squares  of  the  distance  of  the  earth  and  moon  from 
the  sun,  being  a  trifle  less ;  and  2d,  the  quadrants  (so  termed  for 
the  sake  of  conciseness)  nearest  the  sun  contain  a  little  less  than 
90°  each,  and  the  other  two  quadrants,  each  a  little  more  than  90°, 
for  AB  is  not  strictly  a  straight  line,  but  an  arc  of  the  earth's  or- 
bit. A  table  for  two  quadrants  would,  however,  be  sufficient — one 
in  the  half  of  the  orbit  next  to  the  sun,  and  the  other  in  the  half 
most  remote  from  it,  as,  for  example,  DB  and  BC.  Table  12th  is 
constructed  in  this  way,  where  it  will  be  seen  that  the  equations 
are  additive  for  a  little  less  than  90°  after  the  moon  leaves  D,  and 
then  subtractive  to  the  end  of  the  next  quadrant.  If  the  moon's 
angular  distance  from  the  sun  exceeds  180°,  which  would  carry  it 
into  the  quadrants  CA  or  AD,  the  degrees  are  found  at  the  right 
hand  and  bottom  of  the  table,  and  direction  is  given  to  "  reverse 
the  signs,"  so  that  the  equations  which  were  additive  in  DB  become 
subtractive  in  AD,  and  those  which  were  subtractive  in  BC  become 
additive  in  CA.  • 


26 

33.  The  angular  distance  of  the  moon  from  the  sun,  (found  by- 
subtracting  the  longitude  of  the  latter  from  that  of  the  former,  as 
thus  far  corrected,  borrowing  360°  if  necessary,)  shows  in  which 
quadrant  the  moon  is.  When  the  difference  is  from  0°  to  90°,  or 
from  180°  to  270°,  the  moon  is  passing  from  syzygy  to  quadrature, 
but  when  it  is  from  90°  to  180°,  or  from  270°  to  360°,  the  moon  is 
passing  from  quadrature  to  syzygy.  In  the  present  case,  the  lon- 
gitude of  the  sun  (17)  is  64°.3494,  that  of  the  moon  (25)  67°.5374, 
and  the  excess  of  the  latter  2°.  1880.  Entering  table  12th  with 
this  argument,  in  the  same  manner  as  directed  in  article  13th,  the 
equation  is  found  to  be  +.0445.  This  is  to  be  applied  to  the  moon's 
longitude  and  anomaly  according  to  its  sign.  If  the  argument  had 
been  over  180°,  the  sign  of  the  equation  would  have  to  be  changed 
to  — .  After  this  equation  is  applied,  the  moon's  longitude  becomes 
67°.5819,  and  the  anomaly  145°.012. 


34.  The  inequality  to  which  this  chapter  is  devoted,  being  occa- 
sioned by  the  disturbing  influence  of  the  sun,  must  be  more  or  less 
according  as  the  distance  of  that  luminary  varies,  as  we  have  al- 
ready observed,  (31.)  In  table  12th,  and,  consequently,  in  the 
equation  that  was  just  applied,  the  sun  is  supposed  to  be  at  its  mean 
distance.  Hence  another  correction  becomes  necessary,  which 
must  evidently  depend  on  the  same  circumstances  as  the  last,  to- 
gether with  another,  viz.,  the  distance  of  the  sun  from  the  earth, 
which  is  determined  by  its  anomaly.  Accordingly  in  table  13th 
two  arguments  are  employed ;  viz.,  1st,  the  argument  just  used  for 
variation,  which  is  to  be  sought  for  at  the  top  or  bottom  of  the 
table,  and  2d,  the  sun's  anomaly  at  the  right  or  left.  If  the  former 
is  found  at  the  top,  we  look  for  the  latter  at  the  left;  but  if  at  the 
bottom,  at  the  right.  The  equation  is  found  opposite  the  latter, 
and  in  the  same  column  with  the  former.  Since  one  argument  is 
given  only  for  every  5°  and  the  other  for  10°,  it  is  necessary  to 
institute  a  kind  of  double  proportion  for  the  units  and  decimals.  It 
is  further  to  be  noticed,  that  if  both  arguments  are  to  be  found  in 
the  same  gnomon,  enclosed  by  the  heavy  lines  about  the  table,  the 
equation  is  to  be  applied  with  its  proper  sign,  as  found  in  the  table  ; 
but  if  one  is  found  in  the  inner  and  the  other  in  the  outer  gnomon, 
the  sign  before  the  equation  is  to  be  changed  from  +  to  — ,  or 


27 

from  —  to  +.  In  the  present  case,  the  former  argument  (33)  is 
2°.1880,  which  being  between  0°  and  5°,  is  to  be  considered  as 
found  in  the  inner  gnomon  at  the  top;  and  the  latter  (11)  is 
143°.797,  which  is  found  in  the  outer  gnomon,  at  the  left.  Making 
a  proper  allowance  for  the  units  and  decimals,  the  equation  is 
+.0053 ;  but  the  arguments  being  found,  one  in  the  inner  and  the 
other  in  the  outer  gnomon,  the  sign  must  be  changed,  and  the  equa- 
tion becomes  — .0053.  This  applied  to  the  moon's  longitude  and 
anomaly,  found  in  the  last  article,  makes  the  former  C7°.57G6,  and 
the  latter  145°.007. 


CHAPTER  VI. 

PERTURBATIONS  IN  THE  MOON's  MOTION,  CONTINUED. EVECTION. 

35.  The  inequality  which  is  to  occupy  our  attention  in  this  chap- 
ter was  discovered  by  Ptolemy,  A.  D.  110,  and  is  denominated 
Evection. 

For  distinctness  of  conception,  it  is  necessary  to  bear  in  mind 
the  precise  difference  between  this  correction  and  that  treated  of 
in  the  last  chapter,  for  there  is  danger  of  confounding  them,  since 
both  are  caused  by  the  disturbing  force  of  the  sun  in  the  plane  of 
the  ecliptic.  That  supposed  the  original  form  of  the  moon's  orbit 
a  circle,  this  an  elipse,  and  wholly  dependent  on  its  eccentricity  ; 
so  that  if  the  elipse  had  no  eccentricity,  there  would  be  no  correc- 
tion for  evection.  That  always  elongated  the  orbit  in  the  direction 
of  the  quadratures ;  this,  we  shall  see,  elongates  it  in  the  direction 
of  the  syzygyes.  That  regarded  the  shape  of  the  orbit  as  constant ; 
this,  as  ever  changing.  An  important  element  in  this  correction 
is  the  irregular  motion  of  the  line  of  apsides ;  that  had  no  such  line 
to  take  into  account. 


36.  It  will  be  shown,  that  the  progressive  linjjpf  the  moon's  ap- 
sides is  quite  irregular ;  that  it  sometimes  progreSes  more  and  some- 


* 


times  less  rapidly ;  sometimes  remains  stationary,  and  sometimes 
even  goes  backward.  Now  in  determining  the  moon's  mean  ano- 
maly, (11,)  all  the  motions  were  supposed  uniform,  and  no  correc- 
tion has  been  made  for  any  irregularity  in  the  motion  of  the  moon's 
perigee,  except  that  which  resulted  from  the  unequal  distance  of 
the  sun,  (13.)  But  since  the  anomaly  is  reckoned  from  the  peri- 
gee, it  must  be  subject  to  all  the  irregularities  that  the  perigee  itself 
is.  Hence,  in  applying  the  equation  of  the  centre,  (17,)  we  used 
data  that  were  erroneous,  and  the  error  that  was  introduced  needs 
to  be  corrected. 


27.  But  this  is  not  all.  The  greater  the  eccentricity  of  an  orbit 
is,  the  greater  is  the  equation  of  the  centre.  Thus  the  equation  of 
the  moon's  centre  is  much  greater  than  that  of  the  sun  with  the 
same  anomaly,  (compare  tables  8th  and  9th.)  because  the  orbit  of 
the  former  is  much  the  most  eccentric.  Now  it  will  be  shown 
presently,  that  the  eccentricity  of  the  moon's  orbit  is  ever  varying, 
and  the  equation  of  the  centre,  which  depends  upon  it,  must  vary 
likewise ;  whereas  table  9th  is  computed  on  the  supposition  of  a 
constant  mean  eccentricity.  So  that  we  not  only  made  use  of  a 
wrong  anomaly  in  applying  the  equation  of  the  centre,  but  also  a 
wrong  eccentricity  in  the  moon's  orbit.  The  correction  for  the 
combined  effect  of  these  two  errors  constitutes  evection. 


38.  It  will  be  demonstrated  in  its  proper  place,  that  if  a  revolving 
body  be  retained  in  an  eliptical  orbit,  by  a  force  directed  toward  one 
of  the  foci,  the  square  of  the  distance  of  the  body  from  that  focus, 
at  any  point  in  its  orbit,  must  always  be  inversely  proportioned  to 
the  intensity  of  the  attractive  force  at  that  point.  Hence,  if  an  in- 
increase  or  diminution  of  attraction  were  to  take  place  throughout 
the  orbit,  proportional  to  the  existing  attractions  at  each  point, 
the  size,  but  not  the  form  of  the  orbit  would  be  changed.  The  ec- 
centricity would  remain  the  same  as  before.  But  if  the  alteration 
in  the  attractive  force  were  in  any  other  ratio,  it  obviously  would 
affect  the  shape  of  the  orbit.  If  the  perigeal  gravity  was  made  to 
bear  too  great  a  ratio  to  the  apogeal,  it  would  be  drawn  in  too 
much  at  the  former  point,  or  too  little  at  the  latter,  and  the  orbit 
would  become  rrqgfc  eccentric.     Or  if  the  apogeal  gravity  became 


29 

too  great  in  proportion  to  the  perigeal,  the  orbit  would  be  rendered 
less  eccentric,  or  more  nearly  circular. 


39.  To  apply  this  to  our  subject,  let  E  (Fig.  10)  represent  the 
earth,  ADBC  the  moon's  orbit,  A  being  the  perigee  and  B  the  apo- 
gee, and  FGHI  the  sun's  apparent  orbit.  First,  let  the  sun  be  at  S, 
so  that  the  line  of  apsides,  AB,  of  the  moon's  orbit  is  directed  to- 

Fg.  10. 


wards  it,  or  lies  in  syzygy.  The  moon  is  more  attracted  by  the 
earth  at  A  than  it  is  at  B,  in  the  inverse  ratio  of  AE2  to  EB*  ;  and 
in  order  that  the  disturbing  influence  of  the  sun,  which  tends  (28) 
to  diminish  the  earth's  attraction  at  these  points,  should  effect  no 
change  in  the  shape  of  the  moon's  orbit,  it  must  also  (38)  be  more 
at  A  than  at  B,  in  the  same  ratio.  But  instead  of  that,  the  sun's 
disturbing  influence  is  greater  at  B  than  tit  A,  for  the  difference 
between  SE2  and  SB2  is  greater  than  between  SA2  and  SE2  ;  con- 
sequently the  relative  difference  in  the  attractive  forces  at  A  and 
B  toward  E  is  increased,  and  the  orbit  must  become  more  eccen- 


30 

trie.     In  the  same  manner  it  may  be  shown  that  the  eccentricity 
of  the  moon's  orbit  must  be  increased  when  the  sun  is  at  S". 

But  if  the  sun  were  at  S',  and  the  moon  at  A  or  B,  the  latter 
would  be  drawn  toward  the  earth  by  the  sun's  disturbing  influence, 
and  its  gravity  increased,  (28  ;)  but  more  at  B  than  A  in  the  ratio 
of  EB  to  E A,  as  will  appear  if  we  resolve  the  force  in  the  direction 
S'A  into  two  others  in  the  directions  AE  and  ES',  and  that  in  the 
direction  S'B  into  two  in  the  directions  BE  and  ES'.  In  this  case 
the  greatest  addition  is  made  to  the  least  force,  whereas  to  preserve 
the  shape  of  the  orbit  unchanged,  the  additional  gravities  should 
be  in  proportion  to  the  previously  existing  ones.  The  apogeai 
gravity  thus  becomes  too  great  in  proportion  to  the  perigeal,  and 
the  eccentricity  of  the  orbit  is  diminished,  (38.)  We  shall  arrive 
at  the  same  conclusion  if  the  sun  be  supposed  to  be  at  S'". 

Thus  the  eccentricity  of  the  moon's  orbit  is  greatest  when  the 
line  of  its  ipsides  lies  in  syzygy,  and  least  when  it  lies  in  quadrature. 
It  is  plain  that  these  changes  in  eccentricity  occur,  not  instantane- 
ously, but  gradually,  as  the  sun  progresses  in  its  orbit.  The  ec- 
centricity must  diminish  while  the  sun  is  passing  from  S  to  S',  or 
from  S"  to  S'",  and  increase  while  it  is  passing  from  S'  to  S",  or 
from  S'"  to  S,  being  at  its  mean  state  when  the  sun  is  about  half 
way  between  these  points,  as  at  L,  M,  N  and  K.  The  eccentricity 
must  exceed  the  mean  when  it  is  in  the  quadrants  KL,  or  MN,  and 
be  less  than  the  mean  when  it  is  in  the  quadrants  LM  and  NK. 


40.  The  investigation  of  the  irregular  motion  of  the  line  of  the 
apsides  of  the  moon's  orbit,  on  which  the  evection  in  part  depends, 
is  considerably  more  difficult  than  any  of  the  preceding,  and  will 
require  the  reader's  close  attention.  That  it  must,  on  the  whole, 
progress,  will  appear,  when  we  consider  that  the  average  effect  of 
the  sun's  attraction  is  to  draw  the  moon  away  from  the  earth,  and 
thus  to  render  its  orbit  less  curved  than  it  would  otherwise  be. 
Consequently,  after  the  moon  leaves  its  perigee,  or  apogee,  where 
its  motion  is  at  right  angles  with  the  radius  vector,  its  angular  mo- 
tion round  the  earth  must  amount  to  more  than  180°  before  its 
path  will  have  been  deflected  enough  to  intersect  the  radius  vector 
at  right  angles  again.     That  is,  it  must  be  more  than  180°  from 


31  • 

perigee  to  apogee,  or  from  apogee  to  perigee.*  And,  further ; 
since  the  attraction  of  the  sun  sometimes  increases,  and  sometimes 
diminishes  the  moon's  gravity  toward  the  earth,  we  should  con- 
clude the  line  of  its  apsides  must  sometimes  regress  and  sometimes 
progress.  We  must,  however,  go  into  a  more  minute  investiga- 
tion of  this  motion,  to  account  for  all  the  phenomena  to  which  it 
gives  rise.f 


41.  If  the  moon,  or  any  other  body  revolving  in  an  eliptical  orbit, 
should  be  deflected  from  its  natural  course  at  any  point  by  some  dis- 
turbing influence,  so  as  to  move  at  right  angles  to  the  radius  vector, 
the  point  where  such  deflection  occurred  would  thenceforward  become 
one  of  the '  apsides  of  the  orbit,  provided  it  were  not  further  dis- 
turbed. 


This  is  evident,  from  the  fact  that  the  apsides  are  the  only  points 
in  an  eliptical  orbit,  where  the  curve  is  at  right  angles  to  the  radi- 
us vector.  Also  the  body  must  still  revolve  in  an  elipse,  or  some 
other  conic  section,  for  we  shall  demonstrate  hereafter  that  simple 
gravitation  toward  a  fixed  point  can  retain  it  in  no  other  curve. 
Whether  the  point  of  deflection  will  be  the  perigee  or  the  apogee, 
will  depend  on  the  velocity  of  the  time ;  if  it  be  greater  than  the 
mean,  the  point  will  be  the  perigee,  but  if  less,  the  apogee. 


42.  To  apply  this  principle,  let  us  inquire  what  alteration  must 
be  made  in  the  attractive  force  of  the  earth,  E,  (Fig.  11.)  to  bring 
the  motion  of  the  moon  at  right  angles  to  the  radius  vector  at  the 
points  of  its  orbit  C,  D,  F,  and  G,  the  two  former  being  near  the 
perigee,  A,  where  the  velocity  exceeds  the  mean,  and  the  two  lat- 
ter near  apogee,  B,  where  it  is  less.  While  the  moon  is  moving 
from  A,  through  S  to  B,  the  direction  of  its  motion  constantly  makes 
an  obtuse  angle  with  the  radius  vector,  (as  EFM ;)  it  would  be 

*  Playfair. 

t  The  articles  between  this  and  the  56th  may  be  omitted,  if  the  instructer  should  deem 
it  expedient. 


32 


Fig.  11. 


necessary,  therefore,  at  the 
point  F,  that  the  attractive 
force  of  E  should  be  in- 
creased,  to  curve  its  mo- 
tion to  K,  and  thus  bring 
it  at  right  angles  with  EF. 
And,  if  it  were  so  increas- 
ed, the  point  F  would  (44) 
henceforth  become  the  ap- 
ogee, instead  of  B.  In 
other  words,  the  apogee 
would  have  moved  back- 
ward from  B  to  F ;  and  consequently  the  perigee  from  A  to  T, 
for  they  must  always  be  opposite  each  other. 

The  same  reasoning  will  apply  to  the  point  D  ;  yet,  if  the  deflect- 
ing force  should  occur  there,  D  would  become  the  perigee  instead 
of  the  apogee,  on  account  of  the  moon's  greater  velocity  at  that 
point,  (44,)  and  the  apogee  would  be  found  in  the  direction  of  the 
line  DO ;  so  that  the  apsides  would  have  moved  forward  from  A 
to  D,  and  from  B  to  O.  In  the  other  half  of  the  moon's  orbit, 
where  the  direction  of  the  motion  continually  makes  an  acute  angle 
(as  EGR)  with  the  radius  vector  EG  or  EC,  it  is  manifest,  that 
the  attraction  of  E  must  be  diminished,  in  order  to  bring  the  mo- 
tion at  right  angles,  as  GL  and  CH ;  or,  rather,  a  repulsive  force 
must  be  given  to  it.  Such  a  deflection  occurring  at  G,  would 
change  the  place  of  the  apogee  from  B  to  G,  or,  would  make  it 
move  forward.  If  occurring  at  C,  it  would  change  the  place  of 
perigee  from  A  to  C,  or,  would  make  it  move  backward. 


43.  If  instead  of  increasing  the  gravity  at  F,  it  were  diminished, 
it  is  pretty  clear  that  the  apogee  would  move  forward  instead  of 
backward.  To  illustrate  it,  let  us  suppose  the  gravity  to  be  great- 
ly diminished,  so  much  so  as  to  be  nearly  destroyed.  The  moon, 
being  scarcely  attracted  at  all  toward  E,  will  fly  off  nearly  in  a 
tangent  to  the  elipse  at  F,  and  the  apogee  will  be  found  in  that 
direction,  but  infinitely  distant.  If  then,  we  draw  NP  parallel  to 
the  tangent  FM,  it  will  point  to  the  place  of  the  apogee,  which  has, 
therefore,  moved  forward,  equivalent  to  the  arc  from  B  to  V.  If 
the  attraction  were  less  diminished,  it  would  not  move  forward  so 
far,  but  the  reasoning  would  hold  good. 


33 

Similar  reasoning  applied  to  the  points  D,  C,  and  G,  will  show 
that,  by  reversing  the  supposed  alteration  in  the  force  of  gravity 
at  those  points,  we  shall  reverse  also  the  motion  of  the  line  of  the 
apsides. 

Summing  up  our  conclusions,  we  find  that  an  increase  of  the 
moon's  natural  gravity  toward  the  earth,  near  apogee,  on  either 
side,  would  cause  the  line  of  apsides  to  regress ;  while  a  diminution 
would  cause  it  to  progress ;  and  that  the  reverse  takes  place  by  an 
alteration  in  the  natural  force  of  gravity  when  the  moon  is  near  its 
perigee.  I  employ  the  terms  natural  gravity,  and  natural  velocity, 
to  signify  the  gravity  and  velocity  that  the  moon  would  have  if  it 
revolved  regularly  in  its  eliptic  orbit,  undisturbed  by  the  attraction 
of  any  foreign  body. 

44.  It  is  evident  that  an  increase  in  the  moon's  velocity,  and 
consequently  of  its  centrifugal  force,  must  have  nearly  the  same 
effect  as  a  diminution  of  the  earth's  attraction  ;  and  vice  versa. 
Hence,  if  its  velocity  near  apogee  is,  from  any  cause,  rendered 
greater  than  its  natural  velocity  in  that  part  of  its  orbit,  the  line  of 
apsides  must  move  forward,  or  progress ;  and  the  reverse,  if  such 
an  increase  of  velocity  occurs  near  perigee.  If  both  these  causes 
conspire,  (and  we  will  proceed  to  show  that  they  do,)  the  progress 
or  regress  of  the  apsides  must  be  still  more  rapid. 

45.  It  has  been  shown,  (28,)  that  the  sun's  disturbing  influence 
increased  the  moon's  gravity  toward  the  earth  in  quadrature,  but 
diminished  it  in  syzygy ;  and  we  should  suppose  that  there  must 
be  intervening  points,  where  it  exerted  no  influence  either  way. 
These  points  it  is  important  for  us  to  find,  for  when  the  moon  is 
at  these,  the  apsides  of  its  orbit  must  be  at  rest,  so  far  as  their  mo- 
tion is  caused  by  a  variation  in  gravity. 

46.  Let  the  moon  be  at  any  point  M  (Fig.  12)  of  its  orbit,  and 
let  the  sun's  attraction  on  it  at  that  point  be  represented  by  m. 
Resolving  this  force  into  two  others  in  the  directions  ME  and  ES, 
the  proportion  for  the  former  (called  the  addititious  force,  because 
it  increases  the  moon's  gravity  toward  the  earth)  will  read  SM : 
ME  ::m:  the  addititious  force  =gMm=slfcfMEm- 

3 


47.  The  proportion  for  the  latter  force,  in  the  direction  ES,  will 
read  SM  :  ES  : :  ?n  :  the  force  required  =g^-?w.  But  the  earth  is  at- 
tracted by  the  sun  in  the  same  direction,  ES,  and  it  is  the  differ- 
ence of  the  attractions  only  that  exerts  any  disturbing  influence  on 
the  moon  in  this  direction.     We  will  therefore  find  how  much  the 

attraction  of  the  sun  on 
the  earth  is,  and  subtract 
it  from  that  just  found, 
viz.,  —  m.  By  the  laws 
of  gravity  ES2  :  SM2  :  : 
m :  the  sun's  attraction  at 
the  distance  ES.  Hence 
the  earth  is  attracted 
with  a  force  equal  to 
^m,  which  is  to  be  sub- 
tracted from  ^77i.  Re- 
ducing  the  fractions  to  a 
common  denominator, 
and  subtracting,  we  have 
Now  SM= 
very  nearly; 


ES3— SM3 


-m. 


ES3+SM 

ES— EF, 

therefore,  by  involving 
both  sides,  and  rejecting 
the  3d  and  4th  terms  in 
the  right  hand  member 
on  account  of  their  small- 
ness,  we  have  SM3  = 
ES3— 3ES2xEF.  Sub- 
stituting this  value  in  the 
place  of  SM3'  the  above 
fraction,  which  expresses 
the  disturbing  influence 
of  the  sun  in  the  direction 
ES,  becomes  *™l$mm= 
fgm.  Resolving  this 
force  into  two  others, 
one  in  the  direction  EM, 
and  the  other  at  right 
angtes  to  it ;  i.  e.,  in  the 


directions  EG  and  SG,  the  proportion  for  the  former  (called  the 
ablatitious  force,  because  it  diminishes  the  moon's  gravity  toward 


the  earth)  will  read  SE  :  EG,  or  (since  the  triangles  EGS  and  EFM 
are  similar)  ME  :  EF  : :  ^m :  the  ablatitious  force=^~^m. 

48.  The  addititious  and  ablatitious  forces  acting  in  direct  oppo- 
sition, must  neutralize  each  other  at  the  points  in  the  orbit  where 
they  are  equal,  showing  that,  at  such  points,  the  sun's  attraction 
produces  no  effect  on  the  gravity  of  the  moon  toward  the  e«th. 
In  that  part  of  the  orbit  that  lies  between  these  points  and  quadra- 
ture,.there  wilL.be  an  increase  of  gravity,  and  between  those  points 
and  syzygy,  at  Cor  D,  (for  the  demonstration  will  apply  to.  either 
half  of  the  orbit  ACB  or  ADB,)  a  diminution. 

49.  But  since  the  fractions  representing  these  forces  have  a  com- 
mon denominator,  they  will  evidently  be  equal  when  their  numera- 
tors are  equal,  i.  e.,  when  3EP=ME2  ;  or  (extracting  the  square 
root)  when  v/3xEF=ME;  or  (converting  the  equation  into  a 
proportion)  when  ME  :  EF : :  V3  : 1.  But  ME  :  EF : :  1  :  cos. 
MES  ;  therefore,  by  equality  of  ratios,  v/3 : 1 : :  1  :  cos.  MES= 
.5773672,  which  is  the  cosine  of  54°  43'  56".  Hence  the  gravity 
of  the  moon  toward  the  earth  is  diminished  when  it  is  within  54° 
43'  56"  of  syzygy,  and  increased  when  it  is  within  35°  16'  4" -of 
quadrature. 

50.  It  is  worthy  of  notice  here,  that  the  diminution  of  gravity  in 
syzygy  is  about  double  the  increase  in  quadrature.  The  above 
reasoning  shows,  that  the  ablatitious  force  is  to  the  addititious  as 
3EF2  :  ME2-  But  in  syzygy  EF=ME,  and  3EF=3ME2  ;  so  that 
the  difference  between  them  is  2ME2  ;  while,  in  quadrature,  EF 
becomes  0,  and  the  ablatitious  force  disappears,  leaving  the  additi- 
tious force  proportional  to  ME2,  which  is  half  of  2ME2 

51.  There  remains,  not  yet  investigated,  the  tangential  force  in 
the  direction  GS,  or  MH,  one  of  the  parts  into  which  we  resolved 
(47)  that  in  the  direction  ES.  Its  precise  amount  it  is  not  now  ma- 
terial for  us  to  know :  but  it  is  to  be  observed,  that  its  only  influ- 
ence is  to  retard  the  moon's  motion  from  D  to  B  ;  since,  being  at 
right  angles  to  EM,  it  neither  increases  nor  diminishes  the  moon's 
gravity  toward  the  earth.  If  the  moon  was  supposed  to  be  in  any 
of  the  other  quadrants,  and  figures  constructed  on  the  same  princi- 
ple as  this,  we  should  see  that  the  moon  must  be  retarded  in  pass- 


36 

ing  from  D  to  B,  and  from  C  to  A ;  but  accelerated  from  A  to  D, 
and  from  B  to  C.  Hence  its  motion  must  be  swiftest  in  syzygy  at 
C  and  D,  slowest  in  quadrature  at  A  and  B,  and  a  mean  half  way 
between  syzygy  and  quadrature.  This  is  the  same  conclusion  to 
which  we  arrived  by  a  less  rigid  process,  in  article  29th. 

5%  To  show  how  the  motion  of  the  line  of  apsides  is  affected  by 
this  perturbation  in  the  moon's  gravity  and  velocity,  let  us  recur 
again  to  Fig.  10th.  Let  the  sun  be  at  S,  or  S",  so  that  the  line  of 
apsides,  AB,  lies  in  syzygy.  In  this  case,  both  the  moon's  velocity 
in  its  orbit  will  be  increased,  and  its  gravity  towards  the  earth  di- 
minished (48)  at  A  and  B.  Consequently,  (43  and  44,)  the  line  of 
apsides  must  move  forward  when  the  moon  is  near  apogee,  at  B  ; 
but  backward,  when  it  is  near  perigee,  at  A ;  and  if  the  regress 
near  perigee  is  equal  to  the  progress  near  apogee,  they  will  balance 
each  other,  so  as,  on  the  whole,  to  produce  no  change  in  the  posi- 
tion of  the  line  of  apsides.  But  they  are  not  equal,  for  several  rea- 
sons. 

1st.  The  diminution  of  the  moon's  gravity  at  these  points,  and 
the  increase  of  velocity,  are  both  caused  by  the  force  in  the  direc- 
tion ES,  (Fig.  12,)  which  was  obtained  in  article  47,  by  taking  the 
the  difference  of  the  attractions  of  the  sun  upon  the  earth  and 
moon,  in  that  direction.  Consequently,  they  must  depend  on  the 
difference  between  the  distances  of  the  moon  and  earth  from  the 
sun  ;  and  this  difference  is  greater  when  the  moon  is  in  apogee,  than 
when  it  is  in  perigee. 

2d.  The  forces  causing  the  line  of  apsides  to  progress,  act  for  a 
longer  time  than  those  causing  it  to  regress,  because  the  moon  is 
longer  in  describing  the  apogeal  than  the  perigeal  half  of  its  orbit. 

3d.  A  given  force  would  produce  more  effect  on  the  moon  when 
it  is  in  apogee  than  when  it  is  in  perigee,  on  account  of  its  natural 
velocity  being  less  at  the  former  point,  and  therefore  more  easily 
deflected  from  its  orbit.  If  a  cannon  ball  were  moving  but  one 
foot  in  a  second,  it  would  not  be  very  difficult  to  turn  it  out  of  its 
course,  but  not  so  if  it  were  moving  1000  feet  per  second. 

From  all  these  circumstances  combined,  the  line  of  apsides  pro- 
gresses quite  rapidly  when  the  sun  is  at  S,  or  S".  And,  since  the 
same  causes  operate  in  the  same  way,  one  (51)  while  the  sun  is 
passing  through  the  arcs  KL  and  MN,  extending  45°  each  way 
from  S  and  S",  the  direction  of  the  line  of  apsides  of  the  moon's 


57 

orbit,  and  the  other  (49)  through  FG  and  HI,  extending  54°  43'  56" 
from  the  same  points,  it  follows  that  the  line  of  apsides  must  pro- 
gress, at  least,  while  the  sun  is  in  the  quadrants  KL  and  MN. 

53.  When  the  sun  is  in  either  of  the  small  arcs  FK,  LG,  HM  or 
NI,  containing  9°  43'  56"  each,  the  line  of  apsides  is  nearly  station- 
ary :  for  the  perturbations,  both  in  gravity  and  velocity,  being 
near  their  limits  are  very  weak,  and  what  small  force  they  do  ex- 
ert is  in  opposition  to  each  other,  the  former  tending  to  make  the 
apsides  progress,  and  the  latter,  regress. 

54.  Now  let  us  suppose  the  sun  at  S'  or  S'",  so  that  the  line  of 
apsides  shall  be  at  right  angles  to  ES',  or  shall  be  in  quadrature. 
The  moon's  gravity  toward  the  earth,  when  at  A  and  B,  will  now 
be  increased,  (48,)  and  its  velocity  diminished,  (51.)  Consequently, 
(43  and  44,)  the  line  of  apsides  must  progress  when  the  moon  is 
near  perigee,  but  regress  when  it  is  near  apogee.  And,  by  nearly 
the  same  reasoning  as  employed  above,  (52,)  it  may  be  shown,  that 
the  regress  exceeds  the  progress  ;  so  that,  on  the  whole,  the  line 
of  apsides  regresses.  In  like  manner,  it  may  be  shown,  that  it  re- 
gresses, though  less  rapidly,  when  the  sun  is  any  where  in  the  arcs 
IF  or  GH. 

55.  The  regress  here  will,  however,  be  less  rapid  than  the  pro- 
gress when  the  sun  is  in  the  arcs  KL  and  MN,  for  the  perturba- 
tion in  the  moon's  gravity  is  (50)  but  half  as  great.  It  will  also 
be  of  shorter  duration,  for  the  arcs  IF  and  GH  contain  but  70°  32' 
8"  each,  and  the  sun  moving  forward  in  its  orbit  about  1°  per  day, 
while  the  line  of  apsides  moves  backward,  on  an  average,  about 
1-6  of  a  degree  per  day,  each  arc  will  be  passed  over  in  about  61 
days;  when  1  he  line  will  become  nearly  stationary  for  about  10 
days,  (the  time  occupied  in  passing  one  of  the  small  arcs,  as  FK,) 
and  then  begin  to  advance.* 

*  It  may  seem  to  the  reader  erronious,  to  ascribe  any  part  of  the  progress  of  the  moon's 
apsides  to  the  perturbation  in  velocity ;  for,  since  that  is  equal  in  quadrature  and  syzygy, 
and  extends  the  same  distance,  45°,  from  each,  it  would  seem  that,  so  far  as  this  cause 
is  concerned,  the  regress,  when  the  apsides  are  near  quadrature,  should  be  just  equal  to 
the  progress  when  they  are  near  syzygy.  Sir  Isaac  Newton  took  this  view  of  the  subject, 
and  was  greatly  perplexed  at  finding  that  he  could  account  for  but  about  half  the  motion 
of  the  line  of  the  apsides.  The  explanation  here  given,  is,  in  substance,  that  of  Clairaut, 
who  showed  that,  when  the  apsides  regressed,  they  approached  to  meet  the  sun,  thus 
shortening  the  regressive  arc,  and,  consequently,  diminishing  the  perturbation  in  velocity; 
but,  when  they  progressed,  they  receded  from  the  sun,  lengthening  the  progressive  arc, 
and  thereby  increasing  the  perturbation  in  velocity.  So  that  the  perturbation  would  not 
only  be  greater  in  the  latter  case  than  in  the  former,  but  extend  through  a  greater  arc. 


38 

Since  the  line  of  apsides  remains  nearly  stationary  about  40 
days  in  the  year,  moves  backward  about  122,  and  forward  during 
the  remainder,  amounting  to  about  203  days  ;  and  since  its  forward 
motion  is  more  rapid  than  its  backward,  it  is  evident  that,  in  the 
'course  of  a  year,  it  must,  on  the  whole,  advance.  The  rate  of  ad- 
vance is  found  by  observation,  to  be  such  as  to  carry  it  entirely 
round  the  orbit  in  3232  days,  IS  hours,  |P  minutes,  and  29.4  se- 
conds, or  about  nine  years. 

The  foregoing  explanations  have,  I  trust,  made  the  theory  of  the 
variation  in  the  eccentricity  of  the  moon's  orbit,  and  the  irregular 
motion  of  the  line  of  its  apsides  tolerably  clear  to  the  reader.  It 
remains  to  make  a  practical  application  of  it. 

56.  We  have  seen  (39,  52  and  54)  that  the  eccentricity  of  the 
moon's  orbit  exceeds  the  mean,  when  the  sun  is  in  those  quadrants 
where  it  causes  the  line  of  the  moon's  apsides  to  advance,  and  is 
less  than  the  mean  when  the  sun  is  in  the  other  parts  of  its  orbit : 
also,  that  these  changes  occurred  alternately,  and  nearly  in  alter- 
nate quadrants,  the  lines  of  division  being  not  far  from  half  way 
between  syzygy  and  quadrature.  We  will  eneavour  to  represent 
these  changes  by  a  figure,  the  construction  of  which  was  devised 
by  Sir  Isaac  Newton  for  the  purpose,  and  which  observation  shows 
to  be  very  nearly  correct. 

Let  AaBb  (Fig.  13)  represent  the  moon's  eliptical  orbit,  in  its 
mean  state,  M  the  moon,  E  the  earth,  placed  in  one  of  the  foci,  and 
EC  the  mean  eccentricity.  Now,  as  the  eccentricity  varies,  the 
centre  C  will  sometimes  approach  toward  E,  as  far  as  K,  and 
sometimes  recede  from  it  to  I ;  the  distances  CK  and  CI  being  the 
greatest  variation  of  the  eccentricity  from  the  mean.  Describe 
the  circle  IDKH,  and  join  MC  and  ME.  Let  ES  represent  the 
direction  of  the  sun,  and  B  the  moon's  mean  perigee,  i.e.,  the  place 
of  the  perigee  if  it  progressed  uniformly.  Now,  since  it  has  been 
shown  (39)  that  when  ES  is  at  right  angles  to  AB,  the  eccentricity 
is  least,  and  when  it  coincides  with  it,  the  greatest,  it  is  plain  that 
it  must  be  represented  by  EK  in  the  former  case,  and  EI  in  the 
latter.  And  when  ES  is  in  any  other  position,  the  eccentricity 
must  be  represented  by  a  line  longer  than  EK,  and  shorter  than 
EI.  We  shall  effect  this  for  every  possible  position  of  ES,  by  al- 
ways making  the  angular  distance  of  H  from  I,  in  the  direction 
IDKH,  equal  to  twice  the  angle  BES,     The  length  of  EH  will  be 


39 

equal  to  the  eccentricity  at  the  time,  very  nearly.  For  example, 
when  BES=45°,  H  will  coincide  with  D,  making  EH  very  little 
longer  than  EC,  and  thus  showing  that  the  eccentricity  exceeds 
the  mean  in  the  same  small  ratio.  Again,  if  BES=90°,  H  would 
coincide  with  K,  showing  that  the  eccentricity  was  now  a  mini- 
mum, which  agrees  with  what  we  have  already  seen  to  be  true. 
In  the  same  manner  it  may  be  shown,  that  at  any  other  point,  this 
construction  brings  out  very  nearly  the  result  it  should  do  accord- 
ing to  our  previous  reasoning. 

Fig.  13. 


Not  only  will  the  length  of  EH  represent  the  eccentricity  of  the 
moon's  orbit  at  all  times,  but  its  position  will  represent  that  of  the 
line  of  apsides  very  nearly,  never  varying  from  it  more  than  3'. 
Hence,  the  point  H  may  always  be  considered  as  the  centre  of  the 
eclipse ;  thus  changing  the  whole  eclipse  from  the  mean  position 
AaBb,  to  that  represented  by  the  dotted  curve  WXYZ. 


57.  It  was  remarked,  (17.)  that  if  two  lines  were  drawn  from  the 
mean  place  of  the  moon,  one  to  the  centre  of  the  elipse,  and  the 
other  to  the  focus,  round  which  it  revolved,  the  angle  at  the  moon, 


40  , 

contained  by  these  lines,  would  be  equal  to  half  the  equation  of  the 
centre,  very  nearly.  If  the  eccentricity  and  position  of  the  elipse 
remained  unchanged,  the  angle  in  question  would  be  EMC  ;  but 
when,  in  consequence  of  the  change,  H  becomes  the  centre  of  the 
elipse,  the  angle  becomes  EMH.  Therefore  HMC,  which  is  the 
difference  between  these  two  angles,  must  be  half  the  effect  of  the 
disturbing  force  of  the  sun  ;  or,  in  other  words,  it  is  half  the  evec- 
tion  we  have  been  so  long  in  quest  of.  Hence,  if  we  can  find  out 
a  method  of  determining  the  size  of  this  angle,  or  the  conditions  on 
which  its  size  depends,  our  task  is  over,  for  by  doubling  it  we  shall 
have  the  correction  required. 

58.  Draw  Hs  at  right  angles  to  MC.  Then,  since  small  angles 
are  nearly  proportional  to  their  sines,  the  line  H?  must  always  be 
nearly  proportional  to  the  angle  HMS,  and  consequently  to  the 
evection.  But  Hs  is  also  the  sine  of  HCs,  or  its  supplement  HCR  ; 
therefore  the  evection  is  always  proportional  to  the  sine  of  HCR. 
This  angle  we  will  proceed  to  find.  The  angle  SEB=MEB— 
MES ;  therefore  the  angular  distance  of  H  from  I,  in  the  direction 
IDK,  (being,  by  construction,  double  of  SEB,)  =2MEB— 2MES. 
The  angles  MEB  and  MCB  are  nearly  equal,*  the  eccentricity  of 
the  orbit  being  small ;  therefore,  subtracting  MCB,  or  its  equal 
ICR,  from  the  first  member  of  our  equation,  and  MEB  from  the 
last,  we  have  HCR=MEB— 2MES.  Now  MEB  is  the  moon's 
mean  anomaly,  and  MES  is  the  angular  distance  between  the  sun 
and  moon,  or  the  excess  of  the  mean  longitude  of  the  moon  over 
the  true  longitude  of  the  sun.  Hence  the  evection,  which  has 
been  shown  to  be  proportional  to  the  sine  of  HCR,  is  proportional 
to  the  sine  of  the  moon's  mean  anomaly  diminished  by  twice  the 
excess  of  its  mean  longitude  over  the  true  longitude  of  the  sun.f 

*  There  is  danger  that  the  proportions  of  the  different  lines,  as  they  appear  in  the  figure, 
may  mislead  the  reader,  and  it  is  well  to  remember,  that  EC  is  but  about  1-20,  and  KC 
about  1-100  of  CB. 

t  In  this  demonstration,  the  moon's  longitude  is  supposed  to  exceed  that  of  the  sun. 
But  we  shall  arrive  at  the  same  conclusion  if  we  suppose  the  sun's  longitude  the  greatest ; 
as,  for  example,  if  it  be  in  the  direction  ES'.  For,  now,  S'EB=MEB-f  MES' ;  and, 
consequently,  by  the  same  construction  and  reasoning  as  in  the  other  case,  HCR=MEB 
-f-2MES\  But  the  result  is  obviously  the  same,  whether  we  add  the  angle  MES',  or 
subtract  its  supplement ;  that  is,  the  angular  distance  of  M  from  S',  reckoned  in  the  other 
direction,  S'AaBM.  Now,  this  supplementary  distance  is  the  excess  of  the  moon's  lon- 
gitude over  that  of  the  sun,  borrowing  360°,  or  one  revolution  ;  therefore,  the  angle  HCR 
is  still  equal  to  the  moon's  mean  anomaly,  diminished  by  twice  the  excess  of  its  mean 
longitude  over  the  true  longitude  of  the  sun ;  and  the  principle  becomes  general  in  its 
application. 


41 

If  the  reader  will  now  turn  to  table  14th,  he  will  notice  that  this  is 
the  argument  by  which  the  evection  is  taken  out  in  that  table. 

59.  At  the  time  of  our  predicted  eclipse,  the  quantities  are  as 
follows : — 

The  moon's  mean  anomaly  is,  (13,)       -         -         ■         145°.0780* 
The  moon's  mean  longitude  is,  (11,)  64°.2329 

Subtract  sun's  corrected  longitude,  (17,)  65  .3494 

358  .8835x2=357  .7070 


Argument  of  evection, 147.3110 

60.  Entering  table  14th  with  this  number  as  an  argument,  in  the 
same  manner  as  heretofore,  the  required  correction  is  found  to  be 
.7156,  which  the  sign  — ,  placed  at  the  head  of  the  left  hand  col- 
umn, in  which  the  argument  is  in  this  case  found,  shows  to  be 
subtractive.  It  is  plain  also,  from  the  figure,  that  it  should  be  sub- 
tractive;  for,  in  adding  the  equation  of  the  centre,  (17,)  we  added 
twice  the  whole  angle  EMC,  which  was  too  much  by  twice  the 
angle  HMC.  We  now  correct  the  error,  by  subtracting  the  equa- 
tion just  found  from  the  longitude  and  anomaly  previously  obtained, 
(34,)  which  leaves  for  the  former  66°.8610,  and  for  the  latter, 
144°.291. 

61.  All  the  remarks  that  were  made  in  article  34  on  the  subject 
of  variation,  will  apply  also  to  evection,  since  both  are  caused  by 
the  sun's  disturbing  influence.  The  method  of  taking  the  requisite 
correction  from  the  table  (table  15)  is  also  the  same,  only  that  in 
this  case,  the  argument  for  evection,  viz.,  the  moon's  mean  anomaly 
diminished  by  twice  the  excess  of  the  moon's  longitude  over  that 
of  the  sun,  is  to  be  sought  for  at  the  top  or  bottom  of  the  table,  in- 
stead of  the  argument  for  variation. 

The  correction,  as  found  in  the  table,  is  — .0100,  but  since  one 
argument  is  found  in  the  inner  gnomon,  at  the  bottom,  and  the 
other  in  the  outer  one,  at  the  right,  the  sign  is  to  be  changed,  (34,) 
and  the  correction  becomes  -f  .0100,  which,  added  to  the  longitude 
and  anomaly  last  found,  makes  them  respectively  67°.8710,  and 
144°.301. 

*  This  is  the  moon's  mean  anomaly,  corrected  by  the  annual  equation  of  its  perigee, 
which  it  is  proper  to  do,  because  that  inequality  affects  only  the  average  progressive  mo- 
tion of  the  perigee  at  different  seasons  of  the  year,  and  is  in  no  way  connected  with  that 
which  we  are  now  considering,  or  any  other  which  has  reference  to  the  position  of  the 
moon  in  its  orbit 


42 


CHAPTER  VIL 

NODAL    EQUATION    OF  THE    MOON's    LONGITUDE,  AND    REDUCTION  TO  THE 

ECLIPTIC. 

62.  In  the  numerous  corrections  that  we  have  had  occasion  to 
apply  to  the  moon's  longitude  and  anomaly,  growing  out  of  the  dis- 
turbing influence  of  the  sun,  the  orbits  of  both  have  been  supposed 
to  lie  in  the  same  plane  ;  or  the  latter  to  lie  in  the  plane  of  the  or- 
bit of  the  former.  The  first  of  these  suppositions  is  never  true, 
and  the  latter  only  twice  in  a  year ;  viz.,  when  the  sun  passes  the 
moon's  nodes.  At  all  other  times,  it  is  either  on  one  side  of  the 
plane  of  the  moon's  orbit  or  the  other.  Now  it  is  evident  that  the 
sun's  disturbing  influence,  in  the  various  ways  we  have  been  speak- 
ing of,  must  be  less  than  if  it  lay  in  the  plane  of  the  moon's  orbit ; 
for*  in  order  to  make  our  reasoning  good,  its  attraction  must  be 
resolved  into  two  forces,  one  lying  in  the  plane  of  the  orbit,  and 
the  other  at  right  angles  to  it,  which  necessarily  creates  a  loss- of 
force.  If  it  were  always  at  a  fixed  mean  distance  from  the  plane, 
a  proper  allowance  might  be  made  in  computing  the  inequalities, 
and  the  work  would  thus  be  accurate  without  further  correction. 
In  fact,  the  quantities  in  the  tables  which  we  have  been  using,  were 
calculated  on  that  supposition.  But  since  the  distance  is  variable, 
additional  corrections  are  necessary  for  all  that  we  have  applied 
in  the  three  preceding  chapters.  We  will  select,  as  an  example, 
the  annual  equation  of  the  moon's  longitude,  discussed  in  chapter 
4th,  remarking,  as  we  pass,  that  if  we  were  to  attempt  to  apply  all 
the  corrections  resulting  from  causes  like  that  under  consideration, 
and  from  the  effect  of  one  correction  in  altering  the  argument  from 
which  others  had  been  obtained,  our  task  would  be  endless.  The 
business  is,  at  best,  only  a  series  of  approximations. 

63.  When  the  sun  is  passing  one  of  the  moon's  nodes,  being  in 
the  plane  of  the  orbit,  its  attractive  force  exceeds  the  mean,  so  far 
as  the  circumstance  now  under  consideration  is  concerned,  dilating 
the  moon's  orbit  (19)  and  increasing  the  periodic  time  more  than 
usual.  The  moon  must  therefore  fall  behind  its  mean  place,  and 
continue  to  do  so  more  and  more,  till  the  sun  reaches  its  point  of 
mean  distance,  about  45°  from  the  node.  As  the  sun  continues  to 
recede  from  the  plane  of  the  moon's  orbit,  its  disturbing  influence 


43 

must  grow  less*  allowing  the  moon  to  contract  its  orbit  and  shorten 
its  periodic  time,  till  finally,  when  the  former  is  90°*  from  the  node, 
the  latter  will  have  gained  up  what  it  had  lost,  so  that  its  mean  and 
true  place  will  again  coincide. 

The  reverse  of  all  this  will  take  place  when  the  sun  is  in  the 
next  quadrant.  Its  disturbing  influence  being  a  minimum  at  the 
outset,  the  moon  must  get  ahead  of  its  mean  place ;  and -it  wtfl  not 
lose  what  it  thus  gains  till  the  sun  reaches  the  next  node.  Hence, 
if  the  sun's  longitude  exceeds  that  of  one  of  the  moon's  nodes  by- 
less  than  90°,  something  must  be  subtracted  from  the  longitude  and 
anomaly,  as  already  obtained  ;  but  added,  if  the  excess  is  greater 
than  90°.  Or,  reckoning  from  the  ascending  node,  a  subtractive 
equation  must  be  applied  in  the  1st  and  3d  quadrants,  and  an  addi- 
tive one  in  the  2d  and  4th.  Such  an  equation  is  termed  the  Nodal 
Equation  of  the  moon's  longitude.* 

64.  To  find  how  far  the  sun  is  from  the  node,  the  longitude  of 
the  latter  (13)  must  be  subtracted  from  that  of  the  former,  (17.) 
Entering  table  16th  with  the  argument  thus  found,  viz.,  4°.3486,  in 
the  same  manner  as  we  did  table  6th  and  others,  we  find  the  equa- 
tion to  be  .0026,  which  the  sign  —  at  the  head  of  the  column  con- 
taining the  argument,  as  well  as  our  previous  reasoning,  shows 
must  be  subtractive.  The  resulting  longitude  of  the  moon  becomes 
66°.8684,  and  the  anomaly  144°.298.     , 

65.  The  moon's  anomaly  is  now  altered  considerably,  by  reason 
of  the  various  equations  that  have  been  applied  to  it,  from  what  it 
was  when  we  used  it  to  take  out  the  equation  of  the  centre,  in  arti- 
cle 17th  ;  and  since  this  equation  is  a  very  important  one,  our  work 
will  be  more  accurate  if  we  now  take  it  out  again,  and  by  what- 
ever amount  it  differs  from  what  it  was  as  first  taken  out,  correct 
the  moon's  longitude.  The  anomaly,  as  used  in  article  17th,  was 
145°.078,  which  gave  as  an  equation  +3°.4154,  while  now  it  is 
but  144°.298,  which  gives  for  the  equation  -f-3°.4834,  so  that  we 
did  not  add  enough  to  the  moon's  longitade  by  0°.0680.  Adding 
this  now,  we  have  66°.0364,  which  may  be  regarded  as  the  true 
longitude  of  the  moon,  reckoned  on  its  orbit,  or,  as  it  is  usually 
termed,  the  true  Orbit  Longitude. 

*  I  find  no  name  for  this  equation  in  any  treatise  on  astronomy  that  I  have  met  with, 
and  have  given  it  one  that  seems  to  be  indicative  of  its  character. 


44 

66.  The  plane  of  the  lunar  orbit  being  inclined  to  that  of  the 
ecliptic,  causes  longitudes  reckoned  on  it  to  be  different  from  what 
they  would  be  if  reckoned  on  the  ecliptic.  And  since  the  longitudes 
of  the  heavenly  bodies  are  referred  to  the  latter,  the  orbit  longitude 
just  found  needs  one  more  correction  to  reduce  it  to  the  ecliptic.  The 
argument,  found  by  subtracting  the  longitude  of  the  moon's  node 
from  that  of  the  moon  itself,  is  5°. 9356,  and  the  corresponding 
equation,  obtained  from  table  17th,  is  — .0233.  This  applied, 
leaves  66°.9131  for  the  moon's  true  longitude  from  the  mean  vernal 
equinox. 


CHAPTER  VIII. 


LUNAR,  OR  MENSTRUAL  EQUATION  OF  THE  SUn's    LONGITUDE  AND    NUTA- 
TION. 

67.  It  was  observed  in  article  2d,  that  any  motion  or  change  of 
motion  in  the  earth,  produced  apparently  a  precisely  similar  one 
in  the  sun.  Now,  the  earth,  like  the  moon,  revolves  round  the 
common  centre  of  gravity  of  the  two,  and  is,  therefore,  subject  to 
inequalities  in  this  motion,  the  same  in  kind  as  those  we  have  been 
considering  in  that  of  the  moon,  though  far  less  in  degree,  owing 
to  the  earth's  greater  weight,  and  consequently  close  proximity  to 
the  centre  of  gravity.  These  inequalities,  small  in  themselves,  are 
rendered  vastly  smaller  in  their  effect  upon  the  sun's  apparent  mo- 
tion, by  reason  of  the  great  distance  of  the  latter. 

Fig.  14. 


Let  S  (Fig.  14)  represent  the  sun,  ABF  the  earth,  E  its  centre, 
M  the  moon,  and  C  the  common  centre  of  gravity  between  the 
earth  and  moon,  about  which  both  revolve.     The  distance  from  E 


45 

to  C  is  not  far  from  2970  miles,  or  about  three-fourths  of  the  earth's 
radius. 

It  is  manifest  that  the  longitude  of  the  sun,  as  seen  from  E,  will 
differ  from  its  longitude  as  seen  from  C,  by  the  angle  CSE.  When 
the  angle  MES  is  either  0°  or  180°,  the  angle  CSE  will  disappear, 
and  when  it  is  of  any  other  size,  the  latter  angle  can  be  calculated  ; 
for,  in  the  triangle  CES,  the  two  sides,  CS  and  CE,  and  the  angle 
CES  are  known.  We  assume  here,  that  E  revolves  in  a  circle 
round  C,  keeping  CE  of  uniform  length.  It  is  plain  from  the  dia- 
gram, that  if  the  longitude  of  the  moon  exceeds  that  of  the  sun,  the 
latter  will  be  increased  by  the  angle  CSE ;  but  the  contrary,  if 
the  longitude  of  the  sun  is  greatest.  In  other  words,  if  the  longi- 
tude of  the  moon,  diminished  by  that  of  the  sun,  is  less  than  180°, 
the  equation  will  be  additive,  but  if  greater,  subtractive. 

» 

68.  The  longitude  of  the  sun,  as  found  in  chapter  3d,  is  65°.3494, 
and  that  of  the  moon,  as  finally  corrected,  (67,)  66°.9131.  The 
excess  of  the  latter  above  the  former  is  1°.5637.  Entering  table 
18th  with  this  number,  in  the  usual  way,  we  find  that,  in  the  pres- 
ent case,  the  correction  is  inappreciable,  unless  we  extend  our  de- 
cimals further ;  so  that  the  longitude  obtained  in  chapter  3d  is  to 
be  considered  correct. 


69.  The  error  occasioned  by  regarding  the  orbit  of  the  earth's 
centre  as  a  circle  instead  x>f  an  elipse,  as  it  in  fact  is,  might  be  cor- 
rected by  introducing  an  equation  corresponding  to  the  equation  of 
the  centre  of  the  sun  or  moon.  But  it  would  never  be  necessary, 
unless  extreme  accuracy  were  required  ;  for  the  whole  correction 
which  we  just  undertook  to  apply,  when  a  maximum,  is  but  the 
decimals  of  a  degree,  .0021  ;  and  since  the  eccentricity  of  the  or- 
bit amounts  to  hardly  more  than  1-20  of  the  radius,  the  error  can 
never  be  more  than  about  .0001,  which  is  less  than  half  a  second. 
Much  less  then  must  the  inequalities  in  the  motion  be  appreciable. 


46 
CHAPTER  IX.        ■ 

NUTATION  IN  LONGITUDE. 

.70.  The  equinoxes  are  not  stationary,  but  move  slowly,  west- 
ward, which  necessarily  affects  the  longitudes  of  all  the  heavenly 
bodies,  since  they  are  reckoned  from  the  vernal  'equinox.  If  the 
rate  of  motion  were  uniform,  the  longitudes  of  the  sun,  moon,  and 
moon's  nodes,  which  we  have  obtained  in  the  preceding  chapters, 
would  nevertheless  be  correct ;  for  the  tables  from  which  we  ob- 
tained the  mean  longitudes  in  article  11th,  are  based  upon  the  sup- 
position of  a  uniform  rate  of  precession  of  the  equinoxes,  and  allow- 
ance  is  consequently  made  for  it.  But  it  is  not  uniform,  and  we 
are  now  to  look  into  the  causes  of  the  inequality,  and  make  the 
requisite  correction  in  the  longitudes  on  account  of  it. 

71.  If.  a  body  were  to  revolve  round  the  earth,  in  an  orbit  not 
coinciding  with  the  ecliptic,  so  that  it  would  be  sometimes. north  and 
sometimes  south  of  the  plane  of  the  latter,  we  can  see  that  whenever  it 
were  thus  situated,  the  sun's  attraction  must  tend  to  draw  it  back  into 
the  aforesaid  plane.  To  illustrate  by  a  diagram,  let  SD  (Fig.  1 5)  rep- 
resent the  plane  of  the  ecliptic,  and  MM'  that  of  the  revolving 
body,  both  seen  edgewise,*  S  the  sun,  and  E  the  earth.  Fig.  15. 
When  the  body  is  at  M  the  sun's  attration  on  it,  in  the  di- 
rection SM,  may  be  resolved  into  t^o  other  forces,  in  the 
directions  SC  and  CM,  the  latter  of  which  tends  to  draw 
the  body  directly  into  the  plane  SD.  In  like  manner,  when 
the  body  is  at  M',  it  is  drawn  toward  the  plane  SD,  by  a 
force  represented  by  M'D  ;  and  so  of  any  other  point  out 
of  the  plane  of  the  ecliptic.  The  consequence  is,  that  the« 
body,  as  it  revolves  round  its  orbit,  which  we  will  suppose 
it  to  do  in  an  easterly  direction,  is  drawn  into  the  plane  of 
the  ecliptic,  and  made  to  cross  it  sooner,  that  is,  farther 
westward,  every  succeeding  revolution.*  The  same  would 
be  true  of  any  number  of  bodies  similarly  situated,  so  that 
our  reasoning  will  be  good,  even  if  they  were  multiplied  to  such  a 

*  The  reader  will  perceive,  that  the  condition  of  our  supposed  body  corresponds,  in 
every  respect,  with  that  of  the  moon  revolving  round  the  earth,  and  hence  will  see  the 
cause  of  the  retrograde  motion  of  the  moon's  nodes. 


47 

degree  as  to  form  a  continuous  ring  entirely  round  the  orbit.  Nor 
will  it  alter  the  principle,  if  we  suppose  the  bodies,  or  ring,  very 
near  to  the  earth,  or  even  attached  to  it,  only  that,  in  <  the  latter 
case,  they  would  communicate  their  *  motion  to  the  earth,  and  so 
could  not  move  without  dragging  the  earth  with  them,  which 
would  greatly  deaden  any  motion,  or  .change  of  motion  they  would 
otherwise  have. 

Now,  from  the  spheroidal  form  of  the  earth,  there  is  a  protruber- 
ant  mass  of  matter  about  eighteen  miles  thick  girdling  its  equator, 
every  particle  of  which  is  situated  precisely  as  we  supposed  our 
imaginary  bodies  or  ring  to  be.  The  effect  we  have  described 
must  therefore  follow,  and  every  place — as  Quito,  for  example — 
must  every  day,  by  the  diurnal  motion  of  the  earth,  cross  the  eclip- 
tic at  a  point  farther  west  than  on  the  day  previous.  Or,,  to  illus- 
trate farther ;  suppose  the  plane  of  the  ecliptic  to  be  a  vast  sheet 
of  some  material  substance,  with  an  orifice  of  sufficient  size  to  ad- 
mit the  earth,  and  to  allow  it  to  revolve  freely  on  its  axis.  Now, 
if  a  man  were  stationed  on  Mount  Chimborazo,  (which  we  will 
suppose  to  be  on  the  equator,  though  it  is  not  precisely  so,)  and 
every  time  the  earth  rolled  round,  so  as  to  carry  him  under  the 
plane,  which  would  be  every  twelve  hours,  should  mark  on  the 
edge  the  place  under  which  he  passed  it,  these  marks  would  be 
continually  farther  and  farther  west  by  about  fifteen  feet. 


72.  The  attraction  of  the  moon  also  conspires  with  that  of  the 
sun  in  causing  a  precession  of  the  equinoxes  ;  for  the  plane  of  its 
orbit  being  much  more  nearly  coincident  with  the  ecliptic  than  that 
of  the  equator  is,  it  may  be  regarded  as  another  body  lying  in  the 
plane  of  the  ecliptic,  and  conspiring  with  the  sun  in  its  influence 
upon  the  earth. 

73.  It  is  evident  from  Fig.  15th,  that  the  greater  the  inclination 
of  the  planes  SD  and  MM',  the  greater  must  be  the  forces  repre- 
sented by  CM  and  DM',  and  consequently  the  more  rapid  must  be 
the  retrograde  motion  of  the  points  of  intersection.  So  far,  there- 
fore, as  the  moon's  influence  is  concerned,  the  greater  the  obliquity 
of  its  orbit  to  the  equator,  the  more  rapid  must  be  the  precession 
of  the  equinoxes. 


48 

Let  CD  (Fig.  16)  represent 
a  portion  of  the  ecliptic,  seen 
edgewise,  V  the  vernal  equi- 
nox, and  EF  and  GH  portions 
of  the  moon's  orbit,  making  an  ^_ 
angle  of  about  5°  with  the  eclip- 
tic ;  GH  representing  it  when 
the  ascending  node  is  at  V,  and 
EF  when  the  ascending  node 
is  there.  Also,  let  AB  represent  a  portion  of  the  equator,  making 
an  angle  of  about  23*°  with  the  ecliptic.  Then  HVB,  the  inclina- 
tion of  the  moon's  orbit  to  the  equator,  when  the  ascending  node  is 
at  V,  equals  about  28|°  ;  and  FVB,  the  inclination  when  the  as- 
cending node  is  there,  equals  about  18£°. 

74.  When,  therefore,  the  ascending  node,  in  its  retrograde  course, 
passes  the  vernal  equinox,  which  it  does  once  in  about  nineteen 
years,  the  rate  of  precession  must  considerably  exceed  the  mean, 
and  the  equinoxes  must  immediately  get  too  far  west,  which  would 
increase  the  longitude  af  all  the  heavenly  bodies.  The  same  would 
be  true  all  the  while  that  the  node  was  slowly  working  its  way 
backward  round  to  the  autumnal  equinox ;  for  though  the  rate  of 
precession  would  continually  diminish,  and  become  a  mean  when 
the  node  was  90°  back,  or  west  of  the  vernal  equinox,  yet  it  would 
take  the  whole  of  the  next  quadrant  for  it  to  lose  what  it  had  gain- 
ed in  the  first.  Thus,  when  the  ascending  node  gets  round  to  the 
autumnal  equinox,  which  would  bring  the  descending  node  to  the 
vernal,  all  the  longitudes  would  become  right,  or  in  their  mean 
state  again.  But  the  rate  of  precession  is  now  a  minimum,  and  on 
principles  similar  to  those  we  have  been  discussing,  it  is  apparent, 
that,  while  the  node  is  passing  through  the  other  half  of  its  orbit, 
the  longitude  of  all  the  heavenly  bodies  must  be  less  than  the  mean. 
Thus,  for  a  period  of  about  nine  and  a  half  years,  all  longitudes  are 
greater  than  the  mean,  and  then,  for  the  same  period,  less  ;  and  so 
on,  alternately.     This  is  called  Lunar  Nutation  in  Longitude. 

75.  It  is  plain,  that  when  the  ascending  node  is  passing  from  the 
vernal  back  to  the  autumnal  equinox,  its  longitude  must  exceed 
180°,  and  be  less  than  180°  when  it  is  in  the  other  half  of  its  orbit; 
so  that  we  can  know  by  the  longitude  of  the  node,  whether  to  add 
to  or  subtract  from  our  mean  longitudes. 


49 

76.  At  the  time  for  which  we  are  calculating,  the  mean  longi- 
tude of  the  ascending  node  (11)  is  61°0883  ;  and  entering  table 
19th  with  this  argument,  we  find  the  amount  to  be  subtracted  from 
the  longitudes  of  all  the  heavenly  bodies  at  that  time,  is  .0042, 
which  will  leave  us  for  the  moon's  longitude  (G6)  6G°.9089  ;  for  the 
sun's  (18)  G5°.3452,  and  for  the  moon's  node  (13)  60°.9966. 

77.  There  is  another  inequality  in  the  rate  of  the  Precession  of 
the  Equinoxes,  called  Solar  Nutation,  and  occasioned  by  the  vari- 
able distance  of  the  sun  from  the  plane  of  the  equator  in  the  course 
of  a  year.  But  it  is  so  small  (never  amounting  to  much  over  one 
second)  that  it  may  be  disregarded  without  material  error. 


CHAPTER  X. 

TRUE  TIME  LONGITUDES  AXD  ANOMALIES. 

78.  By  comparing  the  true  longitudes  of  the  sun  and  moon,  found 
in  the  last  chapter,  we  find  that  the  latter  is  greatest  by  1°.5637, 
which  shows  that  the  moon  has  passed  by  the  sun,  and  that  the 
eclipse  is  over.  It  remains  (12)  for  us  to  subtract  such  an  amount 
from  the  mean  time  of  new  moon  (11)  as  it  must  have  taken  the 
moon  to  gain  this  difference,  and  in  order  to  do  so,  we  must  know 
the  relative  velocities  of  the  sun  and  moon  in  their  orbits  at  the 
time. 

Their  motions  are  swiftest  in  perigee,  and  grow  slower  as  they 
recede  from  it ;  hence  their  anomalies  are  the  proper  arguments  for 
determining  their  motions.  We  may  therefore  enter  tables  20th 
and  21st,  with  the  anomalies  of  the  sun  and  moon  respectively  as 
arguments,  and  take  out  their  hourly  motions.  The  former  we 
find  to  be  .0401,  and  the  latter  .5018. 

All  the  other  inequalities  treated  of  in  the  preceding  chapters, 
must  likewise  affect  the  moon's  hourly  motion,  of  which  Variation 
and  Evection  are  the  most  important.  The  effect  of  Variation,  as 
is  plain  from  the  theory,  is  to  increase  the  moon's  velocity  in  syzy- 
gy  and  diminish  it  in  quadrature.  Now,  in  an  eclipse,  the  moon  is 
always  in  syzygy,  and  hence  we  must  add  to  its  hourly  motion 
the  quantity  given  in  the  margin  of  table  21st.     To  correct  the 

4 


50 

moon's  hourly  motion  for  Evection,  we  must  enter  table  22d  with 
the  same  argument  that  was  used  for  that  inequality  in  article  GOth, 
and  in  the  middle  column  we  find  the  equation,  which  is  to  be  ap- 
plied to  the  hourly  motion  according  to  its  sign. 

The  following  is  the  operation  in  the  case  before  us : — 
Moon's  hourly  motion,  by  table  21st,        -         -         .5018 
Add  for  Variation, .0115 

.5133 

Subtract  for  Evection,  by  table  22d,        -        -        .0092 


.5041 

Subtract  sun's  hourly  motion,  -  .0401 


Hourly  gain  of  the  moon  upon  the  sun,     -         -        .4640 

Now,  by  simple  proportion,  we  can  find  how  long  it  must  have 
taken  the  moon  to  gain  the  difference  in  the  longitudes  of  the  sun 
and  moon,  viz.  1°.5637.     Thus, 

.4640 :  1  hour  : :  1°.5637  :  the  time  required,  which  is  thus  found 
to  be  3  hours,  22  minutes,  and  12  seconds.  This  subtracted  from 
the  time  of  mean  new  moon,  found  in  article  11th,  leaves  for  the 
true  time  of  new  moon  in  May,  26d.  8h.  48m.  44sec. 

79.  The  time  thus  found  is  Greenwich  time,  and  to  reduce  it  to 
that  of  any  other  place,  allowance  must  be  made  for  the  difference 
of  longitude,  viz.,  4  minutes  of  time  for  each  degree  of  longitude. 
It  is  also  to  be  observed,  that  the  astronomical  day  begins  at  noon, 
and  counts  the  24  hours  round  to  the  next  noon. 

80.  The  longitudes  and  anomalies  of  the  sun  and  moon  must 
now  be  corrected,  by  subtracting  their  motions  during  the  correc- 
tion just  applied  to  the  time.  If  that  correction  had  been  additive, 
this  would  be  so  also.  The  amount  can  be  easily  found  from  their 
hourly  motions,  thus — 

One  hour  :  the  moon's  hourly  motion,  viz.,  .5041  : :  3h.  22m.  12 
sec. :  the  correction  required  in  its  longitude  and  anomaly,  which 
is  thus  found  to  be  1°.6988. 

One  hour :  the  sun's  hourly  motion,  viz.,  .0401  : :  3h.  22m.  12 
sec. :  the  correction  required  in  its  longitude  and  anomaly,  which 
is  thus  found  to  be  0M351. 


51 


At  the  same  time  the  longitude  of  the  node  must  be  corrected, 
by  taking  from  table  4th  its  motion  during  the  same  time,  and  ap- 
plying it,  with  the  contrary  sign  from  that  of  the  other  motions, 
because  it  moves  in  the  opposite  direction. 

81.  The  reader  will  get  a  clearer  idea  of  the  process  of  calcula- 
ting the  time  of  an  eclipse,  if  we  now  give  a  synopsis  of  the 
work  that  we  have  been  through  in  the  foregoing  chapters. 

EXAMPLE. 
Showing  the  method  of  calculating  the  time  of  a  Solar  Eclipse. 


Time. 

Sun's 

Anom-» 

aly. 

Sun's 
Longi- 
tude. 

Moon't 
Anom- 
aly. 

Moon's 
Longi- 
tude. 

Longi- 
tude of 
Node. 

Mean  new  moon,'March,  1854 

d.    h.    m.  s. 
28  10  42  50 
59     1  28     6 

85.586 
58.211 

6.0194 
58.2135 

93.665 
51.634 

6.0194 
58.2135 

64.2158 
-3.1275 

Mean  new  moon  in  May, 

An.  equa'n  moon's  per.  &  node 

26  12  10  56 

143.797 

64.2329 

145.299 
—.221 

64.2329 

61.0883 
—.0875 

+  1.1165 

+3.4154 

Equation  of  the  centre, 

145.078 

61.0008 
—.0042 

An'l  equation  of  moon's  long. 

65.3494 

145.078 
—.112 

67.6483 
—.1118 

Secular  equa.  of  moon's  long., 

144.966 
+.001 

67.53615 
+.0009 

Variation, v 

144.967 
+.045 

67.5374 
+  .0445 

Annual  equation  of  variation, . 

145.012 
—.005 

67.5819 
—.0053 

Evection, 

145.007 
—.716 

67.5766 
—.7156 

Annual  equation  of  evection, . 

144.291 
+.010 

66.8610 
+.0100 

Nodal  equation  of  the  ) 

144.301 
—.003 

66.8710 
—.0026 

moon's  longitude,  $ 

Corrrection  of  the  equa-  ) 

144.298 

66.8684 
+.0680 

tion  of  the  centre,        £  *  * 

Reduction  to  the  ecliptic, 

66.9364 
—.0233 

—.0042 

—1.699 

Lunar  nutation, 

66.9131 
—.0042 

3  22  12 

—.135 

Correct  for  difference  in  ) 
long,  of  sun  &  moon,  $ 

65.3452 
—.1351 

66.9089 
—1.6988 

60.9966 
+.0074 

True  time,  long's  &  anomalies, 

26    8  48  44 

143.662 

65.2101jl42.599 

65.2101 

61.0040 

52 

82.  It  can  hardly  fail  to  suggest  itself  to  the  attentive  reader, 
that  so  great  an  alteration  in  the  time,  as  that  which  was  made  in 
article  78th,  must  in  some  degree  vitiate  the  result  of  our  work. 
The  anomalies,  and  nearly  all  the  arguments  for  the  inequalities 
would  vary  in  the  interim.  It  would  hence  seem  desirable  to  have 
obtained,  if  possible,  some  nearer  approximation  to  the  true  time  at 
the  outset.  The  "  preliminary  equations"*  in  tables  27  and  28  are 
designed  for  this  purpose,  but  the  theory  of  them  could  not  be  well 
explained  at  that  stage  of  our  work,  where  it  was  necessary  to  in- 
troduce them,  if  at  all. 

The  construction  of  these  tables  is  as  follows.  Since  at  the  time 
of  new  or  full  moon,  the  argument  for  evection  is  the  same  as  that 
for  the  equation  of  the  moon's  centre,  viz.,  the  moon's  mean  anom- 
aly, the  separate  effects  of  the  two  on  the  time  are  united  in  the 
first  preliminary  equation.  Also,  since  both  the  equation  of  the 
sun's  centre,  and  the  annual  equation  of  the  moon's  longitude,  de- 
pend on  the  sun's  anomaly,  they  are  united  in  like  manner  in  the 
second  preliminary  equation. 

After  having  found  the  time  of  mean  new  or  full  moon,  as  des- 
cribed in  article  11th,  we  apply  to  it  these  equations,  and  then  take 
from  table  4th  the  mean  motions  in  longitude  and  anomaly  during 
the  time  so  applied.  These  motions  applied  to  the  mean  longitudes 
and  anomalies  at  the  mean  time,  give  the  mean  longitudes  and  an- 
omalies at  the  corrected  time  ;  and  wre  then  proceed  to  calculate 
the  true  longitudes  at  the  corrected  time,  in  the  same  manner  as 
we  have  done  for  the  mean  time  in  the  foregoing  chapters.  We 
will  illustrate,  by  example,  the  method  of  using  these  tables,  and  at 
the  same  time  show  how  to  calculate  a  lunar  eclipse. 

83.  It  will  not  be  necessary  to  give  much  more  than  a  synopsis 
of  the  operation,  as  the  time  of  a  lunar  eclipse  is  calculated  precise- 
ly in  the  same  manner  as  one  of  the  sun,  only  that  the  half  lunation 
in  table  3d,  is  used  in  order  to  give  the  time  of  mean  full  moon, 
and  the  longitudes  of  the  sun  and  moon,  instead  of  being  made  to 
agree,  are  made  to  differ  just  180°. 

Let  us  inquire  whether  there  will  be  an  eclipse  of  the  moon 
when  the  sun  passes  its  ascending  node  in  the  year  1844. 

Turning  to  table  2d,  we  find  that  the  longitude  of  the  ascending 

*  These  are  the  same  as  those  usually  termed  1st  and  2d  equations  of  the  mean  to 
ihe  true  syzygy. 


53 

node  in  March  of  that  year  is  258°,  and  we  therefore  (11)  take 
from  table  3d  such  a  number  of  lunations  as,  added  to  the  half  lu- 
nation at  the  foot  of  the  table,  will  contain  258  days,  or  thereabouts. 
In  8£  lunations  there  are  251  days,  which  is  the  nearest  to  258 
that  we  can  get  from  the  table.  This  will  carry  the  time  forward 
to  November,  and  will  give  us  for  the  longitude  of  the  sun  244°, 
and  of  the  node  about  245°.  The  sun  will,  therefore,  be  but  1° 
from  the  node,  and  so  far  within  the  lunar  ecliptic  limit  (7)  that 
there  cannot  fail  to  be  an  eclipse  of  considerable  size.  We  will 
proceed  to  calculate  it.  From  tables  2d,  3d  and  5th,  we  obtain 
the  following : — 


Time. 

Sun's 
Anom- 
aly. 

Sun's 
Longi- 
tude. 

Moon's 
Anom- 
aly. 

Moon's 
Longi- 
tude. 

Longi- 
tude of 
Node. 

Mean  new  moon  in  March,  1844, 
Add  eight  lunations, 

d.    h.  m.  s. 

18  15  40  54 

246    5  52  23 

14  18  22     1 

76.522 

232.843 

14.553 

356.7831 

232.8530 

14  5534 

132.366 
206  535 
192.908 

356.7831 
232.8530 
194.5534 

25°.  1245 
—12.5102 
—    .7819 

Mean  full  moon  in  November,... 

24  15  55  18 

323.9181  244.1895 

171809 

64.1895 

244.8324 

Entering  tables  27  and  28,  with  the  anomalies  of  the  moon  and 
sun  respectively  as  arguments,  and  making  the  proper  proportions, 
we  obtian  the  "  preliminary  equations,"  the  first  of  which  is  lh. 
30m.  lsec,  subtractive,  and  the  second  2h.  30m.  J9sec,  also  sub- 
tractive.  The  sum  of  the  two  is  4h.  0m.  20sec,  which  must  be 
subtracted  from  the  mean  time  found  above.  Also  the  motions  of 
the  sun  and  moon  during  this  time,  both  in  longitude  and  anomaly, 
must  be  subtracted,  and  that  of  the  node  added,  because  it  moves 
the  other  way.  The  quantities  in  table  4th,  from  which  we  obtain 
these  motions,  are  given  only  for  the  units  of  the  arguments,  but 
will  answer  just  as  well  for  tens,  by  removing  the  decimal  point 
one  place  to  the  right.  Thus  the  sun's  motion  in  anomaly,  accord- 
ing to  the  table,  is,  for  2  days,  1°.9712,  and  for  20  days  19°.712. 
When  used  for  units,  the  right  hand  figure  may  be  omitted,  and  the 
next  reckoned  according  to  its  nearest  value.  The  following  is 
the  operation  for  finding  the  motions  in  the  case  before  us : — 


Motion  in  4  hours, 

Do.     in  20  seconds, 

Total  motion  in  4h.  0m.  20sec, 


Sun's 
Anom- 
aly. 


0°.164 
0°.000 


0°.164 


Sun's 
Longi- 
tude. 


0°.1643 
0°.0002 


0°.1645 


MoonV 
Anom- 
aly. 


Moon's 
Longi- 
tude. 


2°.177  2°.1961 
0°.003i  0°.O030 


2°.180,2°.1991 


Longi- 
tude of 
Node. 


0°.00Pl 
0°.000(  ! 

! 

0°.008^ 


54 

84.  After  applying  these  corrections  to  the  time,  longitudes  and 
anomalies,  the  process  of  calculation  is,  throughout,  the  same  as 
for  a  solar  eclipse,  with  the  exception  already  mentioned*  and  it  is 
unnecessary  to  go  through  it  in  detail.  The  following  is  a  synop- 
sis of  the  calculation  : — 


EXAMPLE. 

Showing  the  method  of  calculating  the  time  of  a  Lunar  Eclipse. 


Time. 

Sun's 
Anom- 
aly. 

Sun's 
Longi- 
tude. 

Moon's 
Anom- 
aly. 

Moen's 
Longi- 
tude. 

Longi- 
tude of 
Node. 

Mean  new  moon  in  March,  1844, 

d.    h.  m.  s. 
18 15  40  54 
136    5  52  23 
14  18  22    1 

7°6.522 

356.7831 

132.366 

356.7831 

25°3.1245 

14.553    14.5534 

192  908  lOAjaan 

-12.5102 
—  .7819 
244.8324 
+.0088 

64.1895 

24  15  55  18 
—    4    0  20 

323.918  244.1895 
—  .164:    —.1645 

171.809 

1st  cor.  in  time,  with  cor'pond'g  motions, 

—2.180—2.1991 

Annual  equation  of  moon's  per.  and  node, 

24  11  54  58 

323.754  244.0250 

169.629 
+  .217 

61.9904 
+1.0412 

244.8412 
+.0897 

—1.1560 

169.846 

+  .109 

169.955 
+  .001 

169.956 
+  .006 

244.9309 
+.0044 

242.8690 

63.0316 
+.1087 

63.1403 
+.0006 

63.1409 

+.0058 

169.902 
+  .003 

169.965 
—  .193 

63.1467 
+.0034 

63.1501 
—.1930 

169.772 
+  .007 

62.9571 
+.0068 

169.779 
+  .001 

169.780 

62.9639 
+.0013 

62.9652 

+.0063 

62.9720 
+.0077 

+  .0044 

242.8734 
—  .0103 

—  .121 

62.9797 

+.0044 

—        14  41 

—  ".010 

Cor.  for  difference*  in  Ion.  of  sun  &  moon, 

02.984 1 
—.1210 

244.9353 
+.0004 

7-010  time,  longitudes  and  anomalies, 

24  11  40  17 

323.744  242.8631 

169.659 

62.8631 

244.9357 

*  In  lunar  eclipses  it  is  this  difference,  +  or  —180°. 


55 
CHAPTER  XL 

ELEMENTS  OF  AN  ECLIPSE. 

85.  The  following  elements  or  data,  are  all  that  are  needed  for 
making  any  calculation  that  we  may  desire  in  regard  to  an  eclipse, 
either  solar  or  lunar ;  such  as  the  place  on  the  earth's  surface, 
where  the  sun  will  be  centrally  eclipsed  at  any  given  time,  while 
the  eclipse  lasts ;  the  portions  of  the  earth  where  an  eclipse  will 
be  visible,  and  the  time  when  it  will  commence,  become  a  maxi- 
mum, and  terminate  ;  or  the  size  of  an  eclipse,  at  any  given  place 
and  time.  The  10th  is  not  needed  in  solar  eclipses,  nor  the  2d,  3d, 
and  12th  in  lunar. 

1st.  The  time  of  new  or  full  moon. 

2d.  The  longitudes  of  the  sun  and  moon. 

3d.  The  obliquity  of  the  ecliptic  to  the  equator. 

4th.  The  moon's  latitude. 

5th.  The  sun's  hourly  motion. 

6th.  The  moon's  relative  hourly  motion,  or  the  excess  of  its 
hourly  motion  over  that  of  the  sun. 

7th.  The  sun's  apparent  semidiameter  as  seen  from  the  earth. 

8th.  The  moon's  do. 

9th.  The  apparent  semidiameter  of  the  earth  as  seen  from  the 
moon,  which  is  the  same  as  the  moon's  horizontal  parallax. 

10th.  The  apparent  semidiameter  of  the  earth's  shadow  where 
it  eclipses  the  moon,  as  seen  from  the  earth. 

11th.  The  angle  of  the  moon's  visible  path  with  the  ecliptic. 

12th.  The  sun's  declination. 

86.  The  method  of  obtaining  the  first  two,  was  explained  in  the 
last  chapter. 

87.  The  mean  obliquity  of  the  ecliptic  to  the  equator  in  the  year 
1840,  was  24°  27'  62".52,  but  it  decreases  at  the  rate  of  about  half 
a  second  a  year,  owing  to  the  attraction  of  the  planets.  Il  is  also 
subject  to  an  inequality,  whose  period  is  about  19  years,  depending 
on  the  longitude  of  the  moon's  node :  for  it  is  evident  from  the 
theory  of  lunar  nutation,  that  the  moon's  influence  must  affect  the 


56 


obliquity  of  the  equator  to  the  ecliptic,  as  well  as  the  place  of  their 
points  of  intersection.  Table  23d  gives  the  obliquity  on  the  1st  of 
January  in  each  year  of  the  present  century  after  1840,  taking  both 
these  causes  into  account,  from  which  it  can  be  readily  found  for 
any  time  in  the  year  by  inspection.  In  January,  1854,  it  is  23°  27' 
33".6,  and  in  January,  1855,  it  is  23°  27'  35".7.  Hence,  at  the 
time  of  our  solar  eclipse  in  May,  1854,  it  is  23°  27'  34".5. 


88.  The  moon's  latitude  depends  on  its  distance  from  the  node, 
and  the  inclination  of  the  plane  of  its  orbit ;  but  the  inclination,  and 
also  the  place  of  the  node,  varies  according  to  the  situation  of  the 
sun  in  respect  to  the  node.  Hence  there  are  two  principal  equa- 
tions of  the  moon's  latitude,  one  depending  on  the  distance  of  the 
moon  from  the  node,  and  the  other  on  that  of  the  sun.  Now  in 
eclipses,  the  distance  of  both  luminaries  from  the  node  is  the  same, 
so  that  the  two  equations  may  be  combined  into  one,  which  is  done 
in  table  24th.  The  argument  is  found  by  subtracting  the  longitude 
of  the  node  from  that  of  the  sun  and  moon,  which  leaves,  in  our 
solar  eclipse,  4°.2061,  and  in  the  lunar,  177°.9274.  The  moon's 
latitude,  as  determined  by  these  arguments,  is,  in  the  former  case, 
the  decimals  of  a  degree  .3664,  or  a  little  over  one-third  of  a  de- 
gree, and  in  the  latter,  .1810.  It  is  to  be  noticed,  that  in  table  24th 
the  figures  of  the  argument  at  the  right  and  left  are  whole  degrees, 
and  those  at  the  top  and  bottom  the  first  place  of  decimals. 

It  must  be  readily  seen,  that  the  moon's  latitude  must  be  north 
for  the  first  180°  after  it  leaves  the  ascending  node  ;  and  that  it 
moves  northerly,  or  ascends,  through  the  first  quadrant,  and  south- 
erly, or  descends,  through  the  second :  also,  that  in  the  other  half 
of  the  orbit,  its  latitude  must  be  south,  being  descending  in  the  first 
quadrant,  and  ascending  in  the  second.  These  facts  are  indicated 
in  the  table  by  capital  letters  at  the  head  of  the  columns  containing 
the  argument. 


89.  The  method  of  finding  the  5th  and  6th  elements  was  explain- 
ed in  the  last  chapter,  (78  :)  but  if  much  accuracy  were  required, 
they  wrould  have  to  be  now  computed  over  again  for  our  solar 
eclipse,  because  the  anomalies  have  been  changed.  Calculated 
from  the  anomalies  as  finally  corrected,  in  the  same  manner  as 
was  done  in  article  78,  the  hourly  motion  of  the  sun  in  our  solar 


57 

eclipse  we  find  to  be  .0401,  and  in  the  lunar  .0421  ;  and  the  rela- 
tive hourly  motion  of  the  moon  is  .4649  in  the  solar  eclipse,  and 
.4523  in  the  lunar. 


90.  The  7th  element  is  obtained  from  the  1st  column  of  table 
26th,  where  it  is  sufficiently  explained. 

91.  Table  21st,  columns  1st  and  3d,  give  the  8th  and  9th  ele- 
ments, so  far  as  they  depend  on  the  elliptical  form  of  the  moon's 
orbit.  But  the  effect  of  Variation  is  to  throw  the  orbit  into  a  kind 
of  oval,  with  its  shortest  diameter  lying  in  syzygy.  From  this 
cause,  the  distance  of  the  moon  from  the  earth  is  less  when  it  is 
new  or  full  than  at  other  times,  which  must  increase  their  apparent 
size  as  viewed  from  each  other.  Hence  the  corrections  in  the 
margin  of  table  21st.  Also  Evection,  by  altering  the  shape  of  the 
moon's  orbit,  affects  its  apparent  size  and  parallax,  so  that  a  far- 
ther correction  becomes  necessary  from  table  22d.  The  following 
shows  the  method  of  obtaining  these  elements  for  the  eclipse  of 
May,  1854  :— 


Semidiarneter. 

Parallax. 

Values  taken  from  table  21st, 

.2482 
-f  .0020 

.9097 
+  .0073 

"Variation, 

Evection, 

.2502 
—  .0023 

.9170 
—  .0080 

True  semidiameter  and  parallax, 

.2479 

.9090 

The  semidiameter  and  parallax  at  the  time  of  the  lunar  eclipse 
in  November,  1844,  obtained  in  the  same  way,  are  .2453  and 
.8989  respectively. 

92.  It  was  shown  in  article  4th,  that  our  10th  element,  viz.,  the 
apparent  semidiameter  of  the  section  of  the  earth's  shadow  that 
eclipses  the  moon,  is  equal  to  the  sum  of  the  parallaxes  of  the  sun 
and  moon,  diminished  by  the  sun's  apparent  semidiameter.  The 
sun's  parallax  may  always  be  put  down  at  .0024,  and  the  methods 
of  obtaining  the  other  data  for  finding  this  element  have  been  al- 
ready explained.  Thus  in  the  lunar  eclipse  which  we  have  taken 
as  an  example, 


58 

The  sun's  parallax  is .0024 

The  moon's  do.,  as  just  found,  is     -         -         -         -         .8989 

.9013 

The  sun's  semidiameter  (see  7th  element)  is    -         -        .2707 

The  semidiameter  of  earth's  shadow  is  -         -         -         .6306 

93.  The  angle  which  the  moon's  path  makes  with  the  eclip- 
tic varies  according  to  the  moon's  distance  from  the  node.  When 
at  the  node,  it  makes  the  same  angle  as  the  planes  of  the  two  or- 
bits; but  when  it  is  90°  from  it,  its  motion  becomes  parallel  to  the 
ecliptic.  But  the  inclination  of  the  planes  also  varies,  depending, 
as  was  remarked  above,  on  the  distance  of  the  sun  from  the  node. 
The  two  influences  may,  however,  be  combined  into  one  at  the 
time  of  an  eclipse,  in  the  same  manner  as  in  table  24th.  And  not 
only  does  the  real  angle  vary  from  both  these  causes,  but  the  ap- 
parent angle  is  increased  by  the  earth's  motion  in  the  same  direc- 
tion ;  and  since  the  rate  of  the  latter,  as  compared  with  the  moon's 
motion,  is  quite  variable,  the  apparent  angle  must  vary  also  from 
this  cause.  All  these  causes  are  taken  into  account  in  table  25th. 
This  table  has  two  arguments,  viz.,  1st,  the  difference  between  the 
hourly  motions  of  the  sun  and  moon,  (for  the  motion  of  the  earth  is 
measured  by  the  apparent  motion  of  the  sun,)  the  first  two  decimal 
places  of  which  are  placed  at  the  top ;  and  2d,  the  distance  of  the 
sun  or  moon  from  the  node,  which  is  placed  at  the  right  and  left. 
In  the  solar  eclipse  we  are  calculating,  the  former  (89)  is  .4649, 
and  the  latter  (found  by  subtracting  the  longitude  of  the  node  from 
that  of  the  moon)  4°.2061.  These  give  the  angle  5°  44'  33", 
ascending.  In  the  same  way  the  angle  at  the  time  of  our  lunar 
eclipse  is  found  to  be  5°  45'  41"  descending. 

94.  The  sun's  declination,  which  is  our  12th  element,  can  easily 
be  computed  from  its  longitude  by  spherical  trigonometry,  since 
the  obliquity  of  the  ecliptic  is  known,  (87 ;)  for  its  longitude,  right 
ascension,  and  declination  form  a  right-angled  spherical  triangle, 
of  which  an  angle  and  one  side  is  known.  Table  26th  gives  the 
declination  calculated  from  the  obliquity  in  the  year  1840,  which 
is  sufficiently  exact  for  our  purpose,  though,  after  a  lapse  of  years, 
it  must  evidently  need  correction.  Entering  this  table,  with  the 
sun's  longitude  at  the  time  of  our  solar  eclipse  as  an  argument,  we 
obtain  the  declination  21°.1871. 


59 

Since  the  sun  starts  northerly  from  the  vernal  equinox,  its  decli- 
nation must  be  north  for  the  first  180°,  and  south  through  the  rest 
of  the  orbit.  This  fact  is  indicated  by  the  capital  letters  at  the 
head  of  the  columns  of  the  argument. 

95.  The  elements  collected  are  as  follows  : — 


1.  True  time  of  the  eclipse, 

2.  Longitude  of  the  sun  and  moon,.... 

3.  Obliquity  of  ecliptic  to  equator, 

4.  Moon's  latitude, 

5.  Sun's  hourly  motion, 

6.  Moon's  relative  do 

7.  Sun's  apparent  semidiameter, 

8.  Moon's  do 

9.  Moon's  horizontal  parallax,, 

10.  Semidiameter  of  earth's  shadow, .... 

1 1.  Angle  of  moon's  visible  path  with  eclip. 

12.  Sun's  declination, 


Solar  Eclipse. 


d.  k.  m.  8. 
May  26  8  48  44 , 
65°.2101 
23°  27'  34".5 

0°.3664  (north) 

0°.0401 

0°.4649 

0°.2635 

0°.2479 

0°.9090 


5°  44'  33"  (ascend.) 
2l°.187l  (north) 


Lunar  Eclipse. 


d.    h.    m.   s. 
N"ov.  24  11  40  17 


0°.1810  (north) 

()°.0421 

0°.4523 

0°.2707 

Q°.2453 

3°.8989 

(J°.6306 

5°  45'  4i"  (descend.) 


CHAPTER  XII. 


DELINEATION  OF  A  SOLAR.  ECLIPSE. 


9G.  To  find  whether  a  solar  eclipse  will  be  visible  at  a  particu- 
lar place,  and  if  so,  its  size  and  general  appearance  there,  it  is  more 
convenient  to  first  reverse  the  order  of  viewing  the  phenomenon, 
and  to  suppose  the  spectator  placed  at  the  centre  of  the  sun,  to 
look  down  upon  the  earth,  and  see  the  moon  passing  across  its  disc. 
From  so  vast  a  distance,  the  earth  wofla1  appear  to  him,  as  the  sun 
does  to  us,  like  a  flat  circular  disc.  The  circle  of  illumination 
would  be  to  our  observer  the  circle  of  the  disc,  and  all  circles 
whose  planes  were  perpendicular  to  this  would  be  seen  edgewise, 
and  would  appear  to  him  like  straight  lines.  Their  arcs  would  seem 
to  be  only  of  the  length  of  the  straight  lines  that  they  would  sub- 
tend, as  viewed  by  him.  Such  circles  as  were  seen  obliquely, 
would  appear  elliptical  in  their  shape.     Let  us  suppose  him  to  take 


60 

his  station  when  the  sun  is  in  the  vernal  equinox,  about  the  21st  of 
March,  and  to  retain  it  for  a  year,  accompanying  the  sun  in  its  ap- 
parent annual  round.  Being  always  in  the  plane  of  the  ecliptic,  it 
would  appear  to  him  like  a  straight  line  dividing  the  earth  into  two 
equal  parts,  one  half  lying  north  and  the  other  south  of  it.  At  first 
also,  being  in  the  plane  of  the  equator,  it  too  would  be  seen  as  a 
straight  line,  as  well  as  all  the  parallels  of  latitude  ;  yet  not  paral- 
lel with  the  ecliptic.  The  west  end  of  the 
equator  would  be  north  of  the  ecliptic 
and  the  east  end  south,  crossing  it  in  the 
centre  at  an  angle  of  about  23^°,  as  in 
Fig.  17,  where  AB  represents  the  eclip- 
tic, CD  the  equator,  PP'  the  earth's  axis, 
P  and  P'  its  poles,  GH  the  axis  of  the 
ecliptic,  and  1  1,  2  2,  3  3,  &c,  parallels 
of  latitude. 

As  the  sun  advances,  it  gets  north  of  the  plane  of  the  equator  and 
of  the  parallels  of  latitude,  and  they  will  no  longer  appear  as 
straight  lines,  but  will  seem  bent  downward  toward  the  south. 
The  earth's  axis  will  approach  to  parallelism  with  that  of  the  eclip- 
tic ;  and  the  poles  revolving  in  circles  whose  planes  are  parallel  to 
that  of  the  ecliptic,  will  seem  to  move  in  straight  lines  toward  n 
and  s  till  on  the  20th  of  June,  or  thereabouts,  when  the  sun  reaches 
the  summer  solstice,  the  two  axes  will  coincide,  and  the  north  pole 
will  be  seen  at  n.  The  south  pole  will  be  invisible,  being  hid  be- 
hind a  segment  of  the  earth  ;  but  if  the  earth  were  transparent  it 
would  appear  at  s. 

97.  The  sun  still  advancing,  the  earth's  axis  will  appear  again 
on  the  other  side  of  GH,  the  poles  will  approach  toward  N  and  S, 
and  the  parallels  of  latitude  will  become  less  curved.  And  when 
the  sun  reaches  the  auturmtal  equinox,  in  September,  the  latter  will 
again  become  straight  lines,  but  lying  the  opposite  way  from  what 
they  did  in  March,  and  the  poles  will  appear  at  N  and  S. 

08.  As  soon  as  the  sun  has  passed  the  autumnal  equinox,  the 
parallels  of  latitude  will  appear  curved  again,  but  upward  toward 
the  north,  instead  of  downward,  because  the  sun  will  now  be  on 
the  south  side  of  their  planes.     The  poles  will  recede  along  the 


61, 

iines  NP  and  SP',  passing  n  and  s  when  the  sun  is  at  the  winter 
solstice,  in  December,  and  finally  arriving  at  P  and  P'  about  the 
'21st  of  March,  when  everything  assumes  the  same  aspect»as  when 
he  started. 


99.  A  common  terrestrial  globe  will  serve  to  present  these  vari- 
ous appearances  to  the  reader's  view,  in  a  much  clearer  light  than 
can  be  done  by  any  verbal  description.  Let  the  north  pole  be 
elevated  about  6620  above  the  point  marked  "  North,"  on  the  wood- 
en horizon,  and  then  the  latter  will  represent  the  ecliptic.  If  now 
the  globe  be  placed  upon  a  table  in  the  centre  of  the  room,  so  that 
the  wooden  horizon  may  be  on  a  level  with  the  eye,  and  the  read- 
er, after  having  found  the  21st  of  March  on  the  horizon,  should 
retire  across  the  room  in  that  direction,  he  will  see  the  globe  pre- 
cisely as  represented  in  Fig.  17.  Let  him  now  pass  slowly  round 
the  room  in  the  order  of  the  months  on  the  horizon,  and  all  the  ap- 
pearances we  have  described  will  be  presented  to  his  view. 


100.  When  he  is  in  the  direction  marked  May  26,  he  will  have 
a  true  representation  of  the  earth,  as  it  would  appear  to  our  obser- 
ver at  the  sun,, at  the  time  of  the  solar  eclipse  we  have  been  calcu- 
lating. We  will  endeavor  to  represent  the  same  by  a  figure,  and 
also  the  appearance  of  the  moon  passing  over  the  earth's  disc.  In 
order  to  give  proper  proportions  to  the  several  parts  of  our  figure, 
we  must  be  able  to  mark  down  their  relative  dimensions.  In  com- 
mon plans  and  drawings  these  are  given  in  miles,  feet,  inches,  or 
some  other  direct  measure  of  length ;  but  in  astronomy  it  is  found 
more  convenient  to  determine  them  by  the  angle  which  they  would 
subtend  when  viewed  from  a  given  distance,  as  we  did  in  the  last 
chapter.  And  this  answers  the  purpose  just  as  well,  provided  the 
distance  be  sufficiently  great,  for  the  apparent  would  be  very  near- 
ly proportional  to  the  real  size  of  the  object.  The  dimensions 
which  respect  the  moon  and  earth,  given  in  the  last  chapter,  are 
the  angles  that  the  objects  would  subtend  at  a  distance  of  about 
237,000  miles,  or  the  distance  between  the  earth  and  moon.  The 
reader  must  not  here  fall  into  the  error  of  supposing  that  our  obser- 
ver has  changed  his  position.  He  is  still  at  the  sun,  and  these  an- 
gles are  given  merely  for  the  purpose  of  determining  the  relative 
sizes  of  the  objects  that  we  wish  to  draw. 


62 


*%; 


*; 


fj,        t)       89       ?t>       u       so       so 


4-0         SO  GO         70        80         so       ica 

3 


^yrft  T  ri^fi^.l'j^iTufir'T  tirrrl^TTliTrj'i^^rr  nn]ijYrlrn-<TnT}7TnTiTi  nfimJuiT 


G3 

101.  Our  first  business  is  to  make  a  scale  SS  (Fig.  18)  of  any 
convenient  length,  and  divide  it  into  100  equal  parts.  This  scale 
may  be  considered  to  be  of  such  length,  that  if  seen  perpendicularly 
at  the  distance  of  the  moon  from  the  earth,  it  would  subtend  an 
angle  of  one  degree,  and  of  course  each  of  the  parts  would  subtend 
.01  of  a  degree.  The  earth's  semidiameter  (95)  seen  at  that  dis- 
tance, subtends  an  angle  of  0°.O090.  If  therefore,  with  this  dis- 
tance, taken  from  our  scale,  as  radius,  (counting  the  two  first  deci- 
mals places,  with  a  proper  proportion  for  the  two  last,)  we  describe 
the  semicircle  AGB,  the  line  AB  may  represent  that  portion  of  the 
plane  of  the  ecliptic  which  intersects  the  earth,  and  the  whole  se- 
micircle AGB,  the  half  of  the  earth's  disc  that  is  seen  north  of  it. 
Or,  which  is  the  same  thing,  AB  may  represent  the  wooden  hori- 
zon of  the  globe  adjusted  as  above  described,  and  AGB  the  half  of 
the  globe  that  is  above  it. 


102.  We  will  next  find  the  position  of  the  north  pole  of  the 
earth,  i.  e.,  the  point  on  the  disc  where  it  would  be  seen.    A  glance 
at  the  globe  will  show  that  it  is  not  at  the  top,  nor  anywhere  in 
the  circumference  of  the  disc.     In  fact,  we  have  shown  (96)  that 
it  must  appear  to  move  in  the  right  line  PN,  leaving  P  about  the 
21st  of  March,  and  arriving  at  n  about  the  20th  of  June.     Hence, 
at  the  time  of  our  eclipse,  it  must  be  between  P  and  n.   Its  precise 
position  we  are  able  to  determine;  for  its  motion  in  the  small  circle, 
which,  seen  edgewise,  is  represented  by  the  straight  line  PN,  is 
just  equal  to  that  of  the  sun  in  longitude.     Consequently  the  num- 
ber of  degrees  that  it  has  moved  from  P  must  be  equal  to  the  sun's 
longitude.     The  plane  of  this  circle  is  evidently  perpendicular  to 
the  surface  of  the  paper,  yet  for  the  purpose  of  bringing  the  gradu- 
ation in  sight,  we  will  suppose  it  to  turn  on  the  diameter  PN,  till 
it  lies  flat  down,  as  PHN.     The  reader  must  not  suppose  that  any 
circle  will  be  seen  in  this  position  by  our  observer ;  it  is  merely 
drawn  so  temporarily,  for  the  purposes  of  measurement.     The 
sun's  longitude  being  65°.2101,  the  north  pole  must  be  that  number 
of  degrees  from  P,  which  would  bring  it  to  T  ;  and  this  point, 
when  the  circle  is  turned  up  again  into  its  place,  edgewise  upon 
the  paper,  would  be  seen  at  C,  which  is  hence  the  position  that  the 
north  pole  must  occupy  on  the  disc  ;  and  the  line  CE  must  repre- 
sent the  northern  half  of  the  earth's  axis.     We  shall  have  no  fur- 
ther use  for  the  temporary  circle  PHN,  as  drawn,  and  may,  if  we 


64 

choose,  erase  it,  since  it  forms  no  part  of  the  representation  that 
we  wished  to  draw.  It  merely  served  to  enable  us  to  find  the  po- 
sition of  the  north  pole  of  the  earth,  which  is  now  effected. 


103.  The  delineation  of  a  parallel  of  latitude  on  the  disc  will 
show  how  the  diurnal  path  of  a  place  would  appear  to  our  obser- 
ver. We  will  select  for  the  purpose  that  of  the  Astronomical  Ob- 
servatory of  Williams'  College,  lat  42°  42'  51",  Ion.  73°  12'  33" 
west  from  Greenwich ;  the  former  of  which,  converted  into  de- 
grees and  decimals  by  the  aid  of  tables  30  and  31,  is  42°.7142.  If 
the  latitude  were  just  equal  to  the  sun's  declination,  the  sun  would 
be  vertical  at  noon,  and  the  Observatory  would  be  seen  precisely 
in  the  centre  of  the  disc  at  E  ;  but  since  it  exceeds  it  by  21°.5271, 
the  Observatory  must  be  seen  that  distance  north  of  the  point 
where  the  sun  is  vertical,  which  when  projected  on  the  disc,  would 
become  the  sine  of  the  arc,  measured  from  E,  on  the  axis  EC.  To 
find  the  length  of  the  sine,  we  may  count  the  degrees  upward  from 
A,  and  draw  the  sine  DD,  the  length  of  which,  when  applied  from 
E  toward  C,  will  reach  to  the  point  12.  This  point  must  therefore 
be  the  apparent  position  of  the  Observatory  at  noon. 

104.  If  the  earth  were  transparent,  it  would  be  seen  at  midnight 
considerably  farther  north,  as  is  evident  from  an  inspection  of  the 
globe.  The  point  antipodal  to  that  at*  which  the  sun  is  vertical,  and 
which  also  would  be  seen  at  E,  is  as  many  degrees  south  of  the 
equator  as  the  sun's  declination  is  north.  Hence  the  distance  of 
the  Observatory  from  this  point  at  midnight,  must  be  equal  to  the 
latitude  of  the  former  added  to  the  sun's  declination,  which  amounts 
to  63°.9013.  This  arc.  like  the  former,  when  projected  on  the  disc, 
will  be  seen  on  the  axis  EC,  equal  only  to  the  length  of  its  sine, 
which  we  may  find  in  the  same  way,  by  counting  the  number  of 
degrees  upward  from  A,  drawing  the  sine,  LL,  and  laying  it  off  on 
the  axis  from  E  to  K.  The  point  K  will  then  represent  the  appa- 
rent place  of  the  Observatory  at  midnight. 

105.  The  line  K-12  will  be  the  shortest  diameter  of  the  ellipse, 
into  which  the  parallel  of  latitude  appears  to  be  thrown  by  being 
seen  obliquely ;  the  point  O,  midway  between  K  and  12,  its  centre, 
and  the  line  606,  drawn  through  O  at  right  angles  to  EC,  its  long- 


65 

est  diameter.  The  lines  06.  not  being  foreshortened  by  being  seen 
obliquely,  will  appear  of  the  full  length  of  the  radius  of  the  parallel, 
which,  we  know,  is  the  cosine  of  the  latitude.  The  complement  of 
the  latitude  of  the  Observatory  is  47°.2858,  and  we  may  find  RJl, 
its  sine,  in  the  same  way  as  we  did  the  others.  Setting  off  the 
distance  RR  each  way  from  O  to  6  and  6,  we  find  the  extremities 
of  the  longest  diameter,  which  must  be  the  points  on  the  disc  where 
the  Observatory  will  be  seen  at  6  o'clock  in  the  morning,  and.  at 
the  same  hour  in  the  evening. 

106.  Its  position  at  any  other  hour  in  the  day  may  be  found  by 
the  following  process.  Draw  two  circles,  6M6  and  F'KV,  one 
on  the  longest  and  the  other  on  the  shortest  diameter  of  the  ellipse, 
and  divide  each  into  24  parts,  in  the  points  7,  8,  9, 10,  &c.  corres- 
ponding to  the  hours  of  the  day.  Through  the  division  points  of 
the  former  circle,  draw  straight  lines  parallel  to  EG,  the  earth's 
axis ;  and  through  those  of  the  latter,  at  right  angles  to  it.  Note 
the  points  where  the  lines  that  pass  through  the  same  hour  on  both 
the  circles  intersect  each  other,  and  through  them  draw  the  ellipti- 
cal curve  seen  in  the  figure.  This  curve  will  represent  the  paral- 
lel of  latitude  that  we  wished  to  delineate,  or  the  path  of  the  Ob- 
servatory over  the  disc*  The  several  points  of  intersection  mark 
its  position  at  the  different  hours.  The  two  last  circles,  with  the 
lines  connected  with  them,  except  the  path  of  the  place,  may  be 
drawn  in  pencil  mark,  that  they  may  be  erased  after  the  latter  is 
drawn,  since  they  are  of  no  further  use. 


107.  The  construction  we  have  just  completed  will  show  us,  if 
we  wish,  the  time  of  sunrise  or  sunset,  by  noticing  at  what  hour 
the  path  of  the  place  cuts  the  circle  of  the  disc.  In  this  case,  it  is 
a  little  before  5  o'clock  in  the  morning,  and  a  little  after  7  in  the 
evening. 


108.  The  moon's  latitude  (95)  is  0°.3664  north.  We  will  there- 
fore take  this  distance  from  the  scale  SS,  and  measure  it  from  E 
toward  G,  which  gives  X  as  the  place  where  the  centre  of  the  moon 

*  The  reader  will  readily  see  that  if  the  sun's,  declination  had  been  as  far  south  as  it  is 
north  in  this  case,  the  points  K  and  12  must  exchange  places,  and  the  curve  representing 
the  path  of  the  Observatory  must  lie  on  the  upper  side  of  606. 


6G 

will  appear  to  our  observer  to  cross  the  line  EC  The  propriety 
of  this  step  will  appear  from  the  following  considerations  : — 

1st.  The  longitude  of  the  moon,  when  new,  is  the  same  as  that 
of  Jhe  sun  viewed  from  the  earth,  or  of  the  earth  viewed  from  the 
sun ;  it  must  therefore  be  seen  in  the  line  EG. 

2d.  Its  latitude  is  north,  therefore  it  must  be  seen  above  the  line 
AB,  and  not  below  it,  as  it  would  be  if  its  latitude  were  south. 

3d.  Its  latitude,  as  given  in  article  95,  is  supposed  to  be  measur- 
ed at  the  same  distance  as  the  other  angles  for  which  our  scale 
was  made,  so  that  the  scale  furnishes  the  proper  measure. 

The  only  error  that  is  to  be  noticed,  is  that  the  moon's  centre, 
as  seen  by  our  observer  projected  on  the  earth's  disc,  would  be  a 
little  farther  from  the  ecliptic  than  its  real  distance,  owing  to  the 
divergence  of  the  visual  line  between  the  moon  and  earth.  Since 
however  the  distance  of  the  sun  is  so  great,  the  lines  drawn  from 
the  observer's  eye  to  the  centres  of  the  moon  and  earth  must  be 
very  nearly  parallel,  and  the  divergence  just  named  so  small,  that  it 
may  be  disregarded. 

109.  At  the  time  of  our  eclipse,  the  moon's  path  makes  an  angle 
of  5°  44'  33"  with  the  ecliptic,  tending  north.  If,  therefore,  we 
draw  Ey,  making  an  angle  of  that  size  with  EB,  and  YZ  par- 
allel to  it,  the  latter  line  will  represent  the  apparent  track  of  the 
moon's  centre  across  the  earth's  disc.  It  passes  X  at  48  minutes 
and  44  seconds  after  8  in  the  evening,  by  Greenwich  time,  (95,) 
which,  by  Williamstown  time,  is  4  minutes  and  6  seconds  before  4 
in  the  afternoon.  The  precise  point  where  it  will  be  at  4  o'clock, 
or  any  other  instant  we  may  choose  to  name,  may  be  found  from 
its  relative  hourly  motion,  viz.,  0°.4649.  Taking  this  distance  from 
our  scale,  SS,  dividing  it  into  12  equal  parts,  and  thus  making  the 
smaller  scale,  ss,  we  have  its  motion  in  5  minutes.  With  the  aid 
of  this  we  can  judge  by  the  eye  how  far  it  would  move  in  4  min- 
utes and  6  seconds,  and  setting  off  this  distance  from  X  toward  Z, 
we  find  its  position  at  4  o'clock.  Its  position  at  the  hours  2,  3,  5, 
6,  &c,  may  now  be  found  by  measuring  off  its  hourly  motion  each 
way  from  4.  The  hourly  divisions  may  now  be  subdivided  at 
pleasure.     In  the  plate  they  are  divided  into  quarter  hours. 

110.  The  appearance  of  the  moon  as  projected  upon  the  earth's 
disc,  may  be  shown  by  taking  its  semidiameter,  0°.2479,  from  the 


67 

scale  SS,  and  with  it  describing  a  cirele,  as  d  b  c ;  selecting  for  a 
centre,  the  point  where  the  moon's  centre  will  be  at  the  time  for 
which  we  wish  to  represent  its  appearance.  The  plate  shows  how 
it  will  appear  at  32  minutes  past  5,  the  time  at  which  its  centre 
erosses  the  diurnal  path  of  the  Observatory,  according  to  our  draw- 
ing. At  that  time  the  Observatory  will  be  at  W,  a  little  more 
than  half  way  from  5  to  G  on  its  diurnal  path,  and  very  nearly  co- 
inciding with  the  moon's  centre.  It  must  therefore  be  invisible  to 
our  observer,  being  hid  behind  the  moon  ;  and  the  same  must  be 
true  of  a  large  tract  of  country  about  it ;  for  although  part  of  the 
moon  has  passed  off  from  the  earth's  disc,  the  remaining  part  cov- 
ers the  parallel  of  latitude  between  the  hours  8  and  7,  and  some- 
what more.  This  will  amount  to  over  60°  of  longitude,  reckoning 
15°  for  each  hour,  which  would  extend  from  the  Rocky  Mountains 
on  the  west,  about  one-third  of  the  way  across  the  Atlantic  on  the 
east.  Of  course  the  inhabitants  of  this  entire  tract  must  be  unable 
to  see  the  centre  of  the  sun  at  the  time  of  which  we  speak,  and  at 
Williams'  College  almost  its  whole  disc  must  be  hid  ;  for  it  has  just 
been  shown  that  the  locality  is  almost  in  .perfect  range  with  the 
centres  of  the  sun  and  moon. 


111.  It  is  not  difficult  to  determine  precisely  What  part  of  the 
disc  will  be  concealed  from  view,  or  eclipsed.  Suppose  a  line  to 
be  drawn  from  the  Observatory  to  the  centre  of  the  sun,  and  a 
point  to  be  taken  in  it  at  the  distance  of  the  moon.  Let  another  line 
be  drawn  through  this  point,  from  one  edge  of  the  sun,  and  contin- 
ued so  as  to  meet  the  earth's  disc,  at  a  considerable  distance,  evi- 
dently, from  the  Observatory.  If  now  the  upper  end  of  this  line 
be  carried  round  the  circumference  of  the  sun,  the  lower  end  would 
describe  a  circle  on  the  earth's  disc,  having  the  Observatory  for 
its  centre.  And  the  line  itself  would  describe  two  similar  cones, 
having  a  common  vertex  near  the  moon,  and  their  bases,  one  upon 
the  sun,  and  the  other  upon  the  abovementioned  circle  on  the  earth's 
disc.  As  seen  from  the  centre  of  the  sun,  this  circle  would  have 
precisely  the  same  situation  in  respect  to  the  moon,  that  the  sun 
would  have  as  seen  from  the  Observatory ;  so  that  if  a  portion  of 
this  circle  be  hid  from  our  observer  at  the  sun,  by  some  intervening 
object,  a  like  portion  of  the  sun  would  be  hid  when  viewed  from 
the  Observatory.  We  wish  then,  to  determine  the  size  and  posi- 
tion of  this  circle  at  32  minutes  past  5,  that  we  may  see  how  large 


68 

a  part  of  it  is  covered  or  concealed  by  the  moon.  As  viewed  from 
the  fixed  point  spoken  of,  near  the  moon,  it  and  the  sun  would  evi- 
dently subtend  the  same  angle.  But  the  sun  would  subtend  very 
nearly  the  same  angle,  whether  seen  from  that  point  or  from  the 
earth,  for  the  relative  distances  are  nearly  the  same.  Now  the 
sun's  semidiameter,  seen  from  the  earth,  at  that  time  subtends  an 
angle  of  0°.2635  ;  therefore  this  circle  must,  at  the  distance  of  the 
moon,  subtend  the  same  angle.  Hence  we  may  take  this  distance 
from  the  scale  SS,  and  with  it  discribe  the  circle  a  b  c,  from  the 
centre  W,  the  position  of  the  Observatory  at  the  time,  and  we  have 
the  circle  in  question ;  showing  that  a  very  slender  crescent  of 
light  will  be  seen  on  the  south  side  of  the  moon. 

The  width  of  this  crescent  is  usually  described,  by  dividing  the 
diameter  of  the  circle  into  12  equal  parts,  called  digits,  and  seeing 
how  many  of  these  it  contains.  In  this  case  it  contains  about  one- 
fourth  of  one  of  these  divisions,  leaving  11  3-4  digits  eclipsed. 

112.  The  eclipse  must  evidently  commence  at  Williams'  College, 
as  soon  as  the  moon  and  the  circle  that  we  last  drew  begin  to  in- 
terfere, which  must  be  as  soon  as  the  distance  between  their  centres 
becomes  less  than  the  sum  of  their  semidiameters.  The  two  semi- 
diameters  added  together  make  0°.5114,  and  we  may  take  this  dis- 
tance from  the  scale  SS,  and  setting  one  foot  of  our  compasses  on  the 
moon's  path,  some  distance  to  the  left  of  W,  and  the  other  on  that 
of  the  Observatory,  move  them  backward  or  forward  till  both  feet 
stand  on  the  same  hour  and  minute,  which  must  be  the  time  when 
the  eclipse  commences.  By  a  similar  operation  at  the  right  hand 
of  W,  the  time  of  the  end  may  be  found. 

The  results,  according  to  our  drawing,  are  as  follows : — 

h.    in. 

Beginning,       -                                    -         -         -  4  15 

Greatest  obscuration, 5  32 

End, 6  38 

Duration, 2  23 

Digits  eclipsed, 11  3-4 

113.  We  may  derive  from  Fig.  18th  a  method,  by  which  the 
size  and  appearance  of  a  solar  eclipse,  at  any  given  time,  and  place 
may  be  calculated  mathematically.  The  tabular  latitudes  and  lon- 
gitudes of  the  sun  and  moon  are  calculated  for  the  centre  of  the 


69 

earth,  and  are  consequently  correct  only  where  the  sun  and  moon 
are  vertical,  or  in  the  zenith.  At  all  other  places  they  would  be 
affected  by  parallax.  If  the  place  were  situated  so  as  to  appear  on 
the  disc  above  or  below  AB,  the  parallax  would  affect  the  latitude, 
and  if  on  the  right  or  left  of  GE,  the  longitude  would  be  affected. 
Thus  V  is  the  place  of  the  Observatory  at  3  o'clock  P.  M.  on  the 
day  of  our  eclipse,  and  E  that  at  which  the  sun  is  vertical  and  the 
moon  nearly  so,  not  differing  from  it  more  than  about  J9.  Hence 
VE  is  the  sine  of  the  zenith  distance,  to  which  the  total  effect  of 
parallax  is  always  proportional.  Draw  VX  at  right  angles  to  AB, 
and  it  will  represent  the  effect  of  parallax  upon  the  latitude  of  the 
sun  and  moon,  and  EX  upon  the  longitude. 

Having  computed  the  elements  (95)  for  the  time  at  which  we 
wish  to  represent  the  appearance  of  the  eclipse,  the  arc  GP,  which 
is  equal  to  the  obliquity  of  the  ecliptic  to  the  equator,  will  be  known, 
and  calling  the  radius  of  the  semicircle  AGB  unity,  Vn  and  E?i  can 
be  found.  Again,  since  P/i,  the  radius  of  the  circle  NHP,  is  now- 
known,  and  also  the  arc  PT,  being  equal  to  the  sun's  longitude,  Cn 
can  be  found.  Then  in  the  right  angled  triangle  EwC,  the  two 
sides  E;i  and  Cn  are  known,  and  we  can  find  the  angle  GEC. 

If  we  let  s  represent  the  sun's  longitude,  and  m  the  obliquity  of 
the  ecliptic  to  the  equator,  then  tan.  GEC=tan.  mxcos.  s. 

The  lines  EK  and  El2  are  the  sines  of  the  sum  and  difference  of 
the  latitude  of  the  place  and  the  sun's  declination,  and  are  there- 
fore known :  hence  EO,  which  is  equal  to  half  the  sum,  can  be 
found. 

V 

The  line  OO  is  known,  being  the  cosine  of  the  latitude,  and  also 
the  arc  MF,  being  the  hour  arc  from  noon,  or  the  interval  between 
noon  and  the  time  to  which  the  calculations  refer  converted  into 
degrees  ;  hence  01,  which  is  the  sine  of  this  arc,  or  its  equal  UV, 
and  also  IF  its  cosine  may  be  found.  IV  is  equal  to  IF  foreshort- 
ened in  the  ratio  of  radius  to  the  sine  of  the  sun's  declination,  and 
is  therefore  known ;  and  subtracting*  it  or  its  equal  OU  from  EO, 
we  shall  have  EU.  Now  in  the  right  angled  triangle  EUV,  the 
sides  EU  and  UV  are  known,  and  we  can  find  the  side  EV  and 
the  angle  UEV. 

*  If  the  sun's  declination  and  the  latitude  of  the  place  are  both  north  or  both  south, 
subtract ;  hut  if  one  is  north  and  the  other  south,  add. 


70 

If  we  let  /  represent  the  latitude  of  the  place,  d  the  sun's  declina- 
tion, and  t  the  hour  arc  from- noon,  then 
UV— sin.  Zxcos.  I 

EU=sin.  Zxcos.  d±*cos.  Zxsin.  d  cos-,  t 
sin.  t 


tan.  UEV= 


cotan.  /xcos.  c?±*sin.  flfxcos.  t 


We  now  have  the  angles  GEO  and  UEV,  which,  added  togeth- 
er and  subtracted  from  90°,  leave  the  angle  VEX.  We  have  also 
the  line  EV.  Hence,  in  the  right  angled  triangle  VEX,  we  are 
enabled  to  find  EX  and  VX.  Since  the  parallax  of  a  heavenly 
body  at  any  altitude  is  equal  to  the  horizontal  parallax  mutiplied 
by  the  sine  of  the  zenith  distance.f  which  in  this  case  is  EV,  it  fol- 
lows that  if  we  multiply  it  by  EX,  we  shall  get  the  effect  of  paral- 
lax on  the  longitude,  and  if  by  VX,  on  the  latitude. 

The  value  of  EV  and  the  angle  UEV  may  be  obtained,  if  pre- 
ferred, by  another  process.  The  co-latitude  of  the  place  of  obser- 
vation, the  co-declination  of  a  heavenly  body  and  its  zenith  distance 
form  a  spherical  triangle,  in  which  the  two  former  parts  are  in 
this  case  known,  and  also  the  included  angle,  being  the  hour  angle 
from  noon.  Hence  the  third  side  can  be  found,  the  sine  of  which 
is  EV,  and  the  remaining  angles.  Now,  by  the  principles  on  which 
Fig.  18  is  constructed,  the  angle  UEV  is  the  same  as  that  opposite 
the  co-latitude  in  the  sperical  triangle,  and  is  therefore  known. 

Having  corrected  the  latitudes  and  longitudes  of  the  sun  and 
moon  for  the  effect  of  parallax,  their  differences  will  form  two  sides 
of  a  right  angled  triangle,  and  the  distance  between  their  apparent 
centres  will  be  the  hypothenuse.  By  comparing  the  latter  with 
the  apparent  semidiameters  of  the  sun  and  moon,  the  size  of  the 
eclipse  can  be  readily  determined. 

Our  8th  element  is  the  moon's  apparent  semidiameter,  as  seen 
from  the  centre  of  the  earth  ;  but  the  distance  of  the  moon  from 
any  place  on  the  earth's  surface  at  which  it  is  visible  (save  when 
it  is  in  the  horizon)  is  less  than  from  the  centre,  which  must  cause 
it  to  subtend  a  greater  angle.  The  augmentation  is  a  maximum 
when  the  moon  is  in  the  zenith,  and  grows  less  when  it  recedes 
from  it ;  hence  the  sine  of  the  zenith  distance,  EV,  is  the  proper 


*  See  note  on  preceding  page. 

f  Olmsted's  Astronomy,  Art.  82.— Herscheirs  do.,  Art.  303.— Norton's  do.,  Art.  98. 
Gunmearc'a  do.,  Chap,  v.,  Art.  5. 


71 

argument  for  determining  the  amount  of  the  augmentation,  and  is 
so  used  in  table  29th. 

We  will  show  the  results  of  such  calculations  as  we  have  been 
describing,  by  applying  them  to  our  eclipse  at  32  minutes  past  5, 
the  time  at  which.it  is  represented  in  Fig.  18th.  After  computing 
the  necessary  elements  for  the  time  and  the  parallax,  as  above  des- 
cribed, we  have  the  following  : — 


Sun. 

Moon. 

Latitude, 

Sun. 

Moon. 

Longitude, 

65°.2744 
.0020 

66°.0189 
.7644 

0°.0000 
.0011 

0°.4369  north 
.4154 

Parallax, 

Parallax, 

Difference, 

65°  .2724 
65°.2545 

0°.0179 

65°.2545 

U°0011  south 
0°.0215  north 

0°.0215  north 

Difference, 

0°.0226 

Making  these  differences  the  legs  of  a  right  angled  triangle,  and 
regarding  them  as  straight  lines,  which  we  very  safely  do,  since 
they  are  so  small,  we  find  the  hypothenuse  to  be  0°.0288,  which  is 
the  apparent  distance  between;  the  centres  of  the  sun  and  moon. 

The  sine  of  the  zenith  distance  is  found  by  the  calculations  to  be 
.95678,  and  the  consequent  augmentation  of  the  moon's  semidiame- 
ter,  taken  from  table  29th,  0°.0007.  The  moon's  apparent  semidi- 
ameter  would  otherwise  be  at  this  time  0°.2478,  and  applying  this 
correction,  it  becomes  0°.2485.  The  sun's  remains  the  same  as 
was  found  in  article  95th,  viz.,  0°.2G35,  and  the  difference  between 
the  two  semidiameters  is  0°.0150.  By  comparing  this  difference 
with  the  distance  between  the  centres,  viz.,  0°.0288,  we  see  that 
the  sun's  disc  must  extend  beyond  the  moon's  on  one  side  by 
0°.0438,  or  about  2'  38",  while  on  the  other  it  would  fall  short  by 
0°.0138 ;  thus  forming  a  slender  crescent  of  light  about  the  moon, 
as  we  have  before  remarked. 


72 


CHAPTER  XIII. 

CENTRAL  TRACK  OF  A  SOLAR  ECLIPSE. 

114.  If  we  examine  the  plate  principally  referred  to  in  the  last 
chapter,  (Fig.  18,)  we  notice  that  the  moon's  track  crosses  the 
path  of  the  Astronomical  Observatory  of  Williams'  College  twice  ; 
once  about  20  minutes  before  4,  and  again  about  32  minutes  past 
5.  In  the  first  instance,  it  crosses  far  west  of  where  the  Observa- 
tory will  be  at  the  time,  and  in  the  second,  a  little  east.  Counting 
every  hour  as  15°  of  longitude,  the  point  where  it  first  crosses  is 
about  70°  west  of  the  Observatory,  which  carries  it  into  the  Paci- 
fic ocean,  not  far  from  Astoria.  It  crosses  the  second  time  at  a 
point  about  3°  east  of  the  Observatory,  which  is  in  the  Atlantic 
ocean,  off  Cape  Ann.  If  we  had  numerous  elliptical  curves  drawn 
to  represent  the  different  parallels  of  latitude,  we  might,  by  a  pro- 
cess analagous  to  the  foregoing,  determine  over  what  countries  of 
the  earth  the  moon's  centre,  or  more  strictly,  the  centre  of  its 
shadow  would  pass,  from  the  time  it  first  struck  the  disc  on  the 
west  side,  till  it  passed  off  on  the  east.  There  is  however  an  ea- 
sier way  of  effecting  this,  by  means  of  a  figure  of  different  con- 
struction, used  in  connection  with  a  terrestrial  globe. 

115.  Let  ANB  (Fig.  10)  represent  the  northern  half  of  the 
earth's  disc,  as  seen  from  the  sun,  YZ  the  moon's  track,  with  the 
hours  of  Greenwich  time  marked  on  it,  AB  a  portion  of  the  eclip- 
tic, as  in  Fig.  18,  in  the  last  chapter,  and  the  curved  lines  seconda- 
ries to  it,  orthographically  projected  at  intervals  of  15°. 

Now  to  adjust  the  globe  so  as  to  correspond  with  this  figure, 
elevate  the  north  pole,  as  directed  in  the  last  chapter,  (99,)  and  at 
the  point  that  answers  to  N  in  the  figure.  90°  above  the  wooden 
horizon,  which  now  represents  the  ecliptic,  or  23^°  from  the  north 
pole  of  the  earth,  screw  on  the  graduated  quadrant  of  altitude. , 
By  swinging  the  other  end  round,  between  the  globe  and  the  in- 
side of  the  wooden  horizon,  it  may  be  made  to  represent  any  of 
the  lines  NA,  N«,  N6,  &c.  Or  we  may  screw  it  on  at  S,  and  thus 
represent  the  other  half  of  the  curves,  which  becomes  necessary 
when  the  moon's  latitude  is  south. 


73 

Fig.  19. 


The  point  E  in  the  figure  corresponds,  in  the  present  case,  to 
that  marked  May  26th  on  the  wooden  horizon,  and  consequently 
the  points  A  and  B  must  be  found  90°  from  it  on  each  side.  If  the 
lower  end  of  the  quadrant  of  altitude  is  brought  to  one  of  these 
points,  we  shall  have  a  representation  of  the  arc  NA  ;  and  if  to  the 
other,  of  NB.  Our  figure  shows  that  the  moon's  track  crosses  the 
former  about  16°  40'  above  A,  and  the  latter  about  29°  above  B. 
|The  graduation  of  the  quadrant  of  altitude  would  readily  show 
where  these  places  are  on  the  globe,  were  it  not  for  its  diurnal 
revolution,  which  we  must  next  take  into  account. 

116.  It  is  evident  that  the  sun  must  always  be  just  rising  at  all 
places  situated  on  the  line  NA,  and  just  setting  at  all  on  the  line 


74 

NB.  Now,  at  the  equator,  the  sun  rises  invariably  at  6  o'clock  in 
the  morning,  and  sets  at  the  same  hour  in  the  evening.  Therefore, 
at  8  minutes  before  7  in  the  evening,  by  Greenwich  time,  when, 
according  to  our  drawing,  the  moon's  centre  first  strikes  the  earth's 
disc,  it  must  be  just  6  o'clock  in  the  morning  at  the  place  where 
the  equator  cuts  the  arc  NA,  (compare  Fig.  17.)  The  question  is 
then  reduced  to  this,  viz.,  at  what  place  on  the  equator  is  it  6 
o'clock  in  the  morning,  at  the  same  time  that  it  is  8  minutes  before 
7  in  the  evening  by  Greenwich  time  ?  Converting  the  time  into 
longitude,  the  point  in  question  is  found  to  be  in  the  Pacific  ocean, 
a  little  southwest  of  Mulgrave's  island,  in  longitude  167°  east  from 
Greenwich. 

Turn  the  globe  on  its  axis  till  this  point  is  brought  under,  the 
quadrant  of  altitude,  (the  latter  being  adjusted  so  as  to  represent  NA,) 
and  count  16°  40'  upward  from  the  wooden  horizon,  and  we  shall 
discover  the  place  where  the  centre  of  the  eclipse  first  strikes  the 
earth.  The  experiment  shows  it  to  be  near  the  Caroline  Islands, 
lat.  7°  north,  and  Ion.  164°  east. 

117.  According  to  our  drawing,  the  centre  of  the  eclipse  leaves 
the  earth  at  29  minutes  past  10  in  the  evening,  by  Greenwich  time, 
but  at  the  point  where  the  equator  cuts  the  arc  NB  it  is  but  6 
o'clock  in  the  evening.  The  longitude  corresponding  to  this  differ- 
ence in  time  is  67{°  west,  which  is  in  the  southern  part  of  Vene- 
zuela, in  South  America.  Now  turning  the  globe  on  its  axis  till 
this  point  is  brought  under  the  graduated  quadrant,  (adjusted  so  as 
to  represent  NB,)  and  counting  upward  29°  from  the  wooden  hori- 
zon, we  find  that  the  eclipse  leaves  the  earth  in  the  Atlantic  ocean, 
about  800  miles  easterly  from  Bermuda. 

118.  Let  it  now  be  required  to  find  where  the  eclipse  is  central 
at  any  time  during  its  passage  across  the  disc;  suppose  at  10 
o'clock  P.  M. 

Through  the  point  of  division  representing  10  o'clock  on  the 
moon's  track,  draw  sjc  parallel  to  AB.  The  arc  Bx  contains  about 
27£°,  which  must  also  be  the  number  of  degrees  in  the  arc  m  10. 
Bring  the  foot  of  the  quadrant  of  altitude  to  m,  which  is  45°  from 
E.  Turn  the  globe  backward  on  its  axis  7i°  from  its  last  posi- 
tion, such  being  the  amount  of  its  motion  between  10  o'clock  and 
29  minutes  past  10,  the  time  for  which  we  last  calculated.     The 


75 

27ith  degree  on  the  quadrant  of  altitude,  reckoned  upward  from 
the  wooden  horizon,  will  mark  where  the  eclipse  must  be  central 
at  10  o'clock.     We  find,  on  trial,  that  iris  near  the  west  end  of 
Lake  Superior,  in  North  America. 
i 

119.  These  mechanical  methods  give  a  tolerable  approximation 
to  the  point  where  a  solar  eclipse  will  be  central  at  any  time ;  and 
by  taking  a  sufficient  number  of  points,  we  may  delineate  its  gene- 
ral track.  But  where  much  accuracy  is  required,  recourse  must 
be  had  to  calculation.  The  theory  of  the  following  process  is  the 
same  as  that  of  the  mechanical  method  just  employed. 


120.  The  first  step  is  to  find  the  time  when  the  centre  of  the 
moon's  shadow  first  strikes  or  leaves  the  earth,*  (I  will  here  adopt 
the  latter,)  and  the  sun's  longitude  at  the  time.  The  calculation  is 
a  very  easy  one,  if  the  moon's  track  across  the  disc  is  considered 
as  a  straight  line,  and  the  earth  as  a  perfect  sphere. 

F^  20.  Let  ASBK  (Fig.  20)  repre- 

sent the  earth's  disc,  and  YZ 
the  moon's  track  across  it. 
In  the  triangle  ELM,  the  two 
sides,  EL  and  EM  are  known, 
the  former  being  the  moon's 
latitude  at  the  time  of  new 
moon,  and  the  latter  its  equa- 
torial parallax.  Also,  the  an- 
gle ELM  is  known,  being 
equal  to  that  of  the  moon's 
path  with  the  ecliptic,  increased  by  90°.  Hence  we  can  find  the 
side  LM,  and  the  angle  LEM,  or  its  complement  MEB.  Knowing 
the  moon's  hourly  motion,  we  can  easily  tell  how  long  it  would 
take  it  to  pass  from  L  to  M,  which,  added  to  the  time  of  new  moon, 
will  give  the  time  of  its  leaving  the  earth. 

The  sun's  motion  in  longitude  during  this  interval  can  be  calcu- 
lated from  its  hourly  motion,  and  thus  its  longitude  at  the  time  the 
eclipse  leaves  the  earth  will  be  known. 

If  still  greater  accuracy  be  required,  the  moon's  latitude,  and 

*  It  is  immaterial  whether  we  take  the  time  when  the  centre  of  the  shadow  strikes  or 
leaves  the  earth,  provided  wc  make  the  other  parts  of  the  process  to  correspond. 


76 

the  difference  in  the  longitudes  of  the  sun  and  moon,  at  the  time 
last  found,  can  be  computed  from  the  tables  ;  the  former  of  which 
is  represented  by  the  line  MP,  and  the  latter  by  EP.  Also  we 
may  compute  again  the  angle  of  the  moon's  path  with  the  ecliptic, 
and  the  horary  motions.  With  these  data  we  could  easily  calcu- 
late the  distance  of  the  point  M  from  the  circumference,  reckoned 
on  the  line  YZ,  and  how  long  it  would  take  the  moon  to  pass 
over  it 


121.  Next,  find  the  longitude  of  the  sun,  and  the  latitude  and 
longitude  of  the  moon  at  the  time  for  which  we  wish  to  calculate 
the  position  of  the  centre  of  the  eclipse.  This  may  be  done  either 
from  their  hourly  motions  and  the  angle  of  the  moon's  path  with 
1he  ecliptic,  or,  if  greater  accuracy  be  required,  directly  from  the 
tables. 

Let  the  sun's  longitude  thus  found  =  s. 

Let  the  moon's  latitude         do.        =  I. 

Let  the  difference  in  the  longitudes  of  the  sun  and  moon  =  d. 

Let  the  sun's  longitude  at  the  time  the  eclipse  leaves  the  earth 

Convert  the  time  to  which  the  calculations  refer  into  degree?, 
minutes  and  seconds,  reckoning  15°  for  an  hour;  subtract  there- 
from 90°,  (borrowing  360°  if  necessary,)  and  let  the  remainder 

Let  the  moon's  equatorial  parallax,  which  may  be  regarded  as 
constant  during  the  eclipse,  =p. 

Let  the  obliquity  of  the  ecliptic  to  the  equator  =m. 

Let  P,  Pi,  Ps,  &c.  =  sundry  arcs  and  angles  obtained  during 
the  process  of  computation. 

Let  the  required  latitude  of  the  centre  of  the  shadow  =  x. 

Let  the  required  longitude,  reckoned  westerly  from  the  meridian 
of  Greenwich,  or  of  that  place  for  which  the  time  is  given,  =  y. 


77 

Then  -=sin.  P, 
P 

d  •        -r> 
=T-:=Sin.  Pi, 

f  p   X  COS.    P 

*S±Pl=P2, 

sin.  mxcos.  P2=sin.  P3, 

tan.  ???xsin.  p2=tan.  P4, 

fsin.  (P±P4)xcos.  P3=sin.  a?=the  latitude, 

cot.  s' 


cos.  m 
tan.  P2 


=tan.  Ps, 
Pe, 


cos.  m 


sin.  P3 


cos.  x 

sin.  #xtan.  P7=Ps, 

jp5 — P6+^±Ps=y=the  longitude. 

The  chief  difficulty  in  applying  these  equations  consists  in  know- 
ing which  of  the  four  possible  values  to  give  to  P,  Pi,  P2,  &c. 
The  following  statements  will  remove  all  doubt. 

P,  Pi,  P3  and  P4  are  each  always  less  than  90°. 

Ps  is  always  of  the  same  affection  as  s'  increased  by  90°. 

P6  is  always  of  the  same  affection  as  P2. 

P7  is  less  or  greater  than  90°,  according  as  P  is  less  or  greater 
than  the  complement  of  P4;  it  never  exceeds  180°. 

Ps  is  always  of  the  same  affection  as  P7. 

122.  It  is  impossible,  by  a  mere  description,  to  convey  to  the 
reader  a  clear  idea  of  the  reason  of  the  several  steps  of  this  pro- 
cess ;  but  if  he  will  take  his  globe,  and  adjust  it  in  the  same  manner 
as  he  would  do  to  find  the  position  of  the  centre  of  the  eclipse  by 
the  previous  mechanical  process,  he  may  be  able  to  discover  suc- 


*  If  after  the  new  moon  -j-  ;  if  before  it  — . 

t  The  sign  before  P4  is  +  if  P2  is  less  than  180°  ;  but  —  if  it  is  greater.  And  in  the 
latter  case  if  P4  is  greater  than  P,  the  latitude  of  the  place  will  be  opposite  in  character 
to  that  of  the  moon ;  i.  e.  if  the  moon's  latitude  is  north  that  of  the  place  will  be  south, 
and  the  contrary. 

X  The  sign  before  Ps  is  -f-  if  Ps  is  between  0°  and  90°,  or  between  270°  and  360°  ; 
but  —  if  P2  is  between  90°  and  270°. 


78 

cessively  the  arcs  and  angles  expressed  byP,  Pi,  &c. ;  and  hence 
to  understand  the  method  by  which  they  are  obtained. 

P  is  the  distance  of  the  centre  of  -the  eclipse  from  the  ecliptic, 
measured  on  a  secondary  to  it  drawn  upon  the  earth's  surface,  as  m 
10,  (Fig.  19.)  Or,  it  is  the  latitude  of  that  point  in  the  heavens,  on 
which  an  observer,  placed  at  the  centre  of  the  earth,  would  see  the 
centre  of  the  shadow  at  the  earth's  surface  projected,  if  the  earth 
were  transparent. 

Pi  is  an  arc  of  the  ecliptic,  intercepted  between  the  aforesaid 
secondary  and  the  point  where  the  sun  is  vertical.  Or,  it  is  the 
difference  between  the  sun's  longitude  and  that  of  the  aforesaid 
point  in  the  heavens. 

Pa  is  the  same  arc  increased  by  the  sun's  longitude.  Or,  it  is  the 
longitude  of  the  aforesaid  point  in  the  heavens. 

P3  is  an  arc  of  &.  great  circle,  drawn  from  the  north  pole  of  the 
equator,  perpendicular  to  the  aforesaid  secondary. 

P4  is  the  arc  of  the  secondary,  intercepted  between  this  perpen- 
dicular and  the  north  pole  of  the  ecliptic. 

Ps  is  the  right  ascension  of  that  point  in  the  equator  where  it  is 
cut  by  a  secondary  to  the  ecliptic  passing  through  the  centre  of 
the  shadow  on  the  earth's  surface,  at  the  time  that,  it  leaves  the 
earth  ;  or,  it  is  the  right  ascension  of  that  point  in  the  heavens  on 
which  the  centre  of  the  shadow  would  be  seen  projected  at  that 
time,  by  an  observer  at  the  centre  of  the  earth. 

P6  is  the  right  ascension  of  that  point  in  the  equator,  where  it  is 
cut  by  the  first  mentioned  secondary. 

P7  is  the  angle  at  the  centre  of  the  shadow,  or  at  the  first  men- 
tioned point  in  the  heavens,  contained  between  secondaries  to  the 
ecliptic  and  equator  passing  through  that  point. 

Ps  is  the  arc  of  the  equator  intercepted  between  these  two  secon- 
daries. 


133.  To  apply  the  process  to  a  particular  case,  let  it  be  required 
to  find  the  place  where  the  solar  eclipse  which  we  have  taken  as 
an  example,  will  be  central  at  20  minutes  and  51  seconds  past  10, 


79 

by  Greenwich  time.     After  the  preparatory  steps  described  in 
articles  120  and  121,  we  have  the  following  data  and  results : — 


DATA. 

RESULTS. 

Time=10£.  20m.  51sec. 

P  =  28° 

58' 

22" 

s'=  65°   16'  50" 

Pi  =  66 

29 

27 

5=65°  16'  34" 

P2=131 

46 

1 

/=  0°.4394 

P3=   15 

22 

40 

d=  0°.7278 

P4=   17 

56 

12 

Z=65°  12'  45" 

a;=  44 

45 

28=the  latitude. 

p=  0°.9072 

Ps=153 

21 

5 

?rc=23°  27'  34".5 

Pg=129 

19 

34 

Pt=  21 

55 

45 

P8==   15 

49 

38 

y=  73 

24 

38—  the  longitude 

Thus  we  find  that  the  centre  of  the  shadow,  at  the  time  just 
mentioned,  is  in  lat.  44°  45'  28",  and  Ion.  73°  24'  38",  which  is  on 
the  west  shore  of  Lake  Champlain,  about  four  miles  north  of  the 
village  of  Piattsburg. 


J24.  In  bringing  to  a  close  our  examination  of  the  solar  eclipse 
of  1854,  it  may  not  be  uninteresting  to  give  a  general  description 
of  it. 

The  centre  of  the  eclipse  first  strikes  the  earth  at  54  minutes 
and  27  seconds  past  6  P.  M.,  (Greenwich  time,)  in  the  Pacific 
ocean,  not  far  from  the  Caroline  Islands,  and  travelling  northeast- 
wardly, nearly  over  the  Sandwich  Islands,  strikes  the  American 
coast  a  little  north  of  Astoria,  in  Oregon  Territory,  about  24  min- 
utes past  9.  Crossing  the  Oregon  Territory,  it  enters  the  British 
possessions,  and  turning  easterly,  and  then  southeasterly,  re-enters 
the  United  States  territories  west  of  Lake  Superior.  At  13  min- 
utes past  10  it  crosses  the  outlet  of  Lake  Superior,  about  100  miles 
N.  W.  from  Michilimackinack.  After  reaching  the  settled  parts 
of  Canada,  it  passes  a  little  south  of  Bytown,  travelling  at  a  rate  of 
70  miles  per  minute,  and  reaches  the  St.  Lawrence  about  20  miles 
below  St.  Regis,  at  20  minutes  past  10.  Again  entering  the  United 
States,  on  the  north  line  of  New  York,  it  arrives  at  the  west  shore 
of  Lake  Champlain,  about  four  miles  north  of  Pittsburgh,  at  20 
minutes  and  51  seconds  past  10.     It  strikes  the  opposite  shore  in 


80 

the  south  part  of  the  town  of  Georgia,  and  from  thence  passes 
through  the  following  towns  in  Vermont,  viz.,  Fairfax,  Fletcher, 
Cambridge,  Stirling,  Morriston,  Elmore,  Woodbury,  Cabot,  Dan- 
ville, Barnet,  Waterford,  and  reaches  the  Connecticut  river  at  21 
minutes  and  45  seconds  past  10.  Travelling  now  about  100  miles 
per  minute,  it  passes  through  the  towns  of  Littleton  and  Bethlehem, 
in  New  Hampshire,  and  from  thence  directly  over  the  Notch  in 
the  White  Mountains,  and  through  Adams  and  Chatham  into 
Maine.  After  passing  through  Fryeburg,  Denmark,  Bridgetown, 
Sebago,  and  across  the  pond  into  Windham,  Gray,  Cumberland, 
and  Yarmouth,  it  strikes  the  Atlantic  in  the  latter  town,  about  ten 
miles,  in  a  direct  line,  from  Portland.  It  leaves  the  earth  at  28 
minutes  and  55  seconds  past  10,  in  lat.  32°  36'  6",  Ion.  49°  44'  12", 
which  is  in  the  Atlantic  ocean,  about  800  miles  east  of  Bermuda.* 

Since  the  apparent  size  of  the  moon  at  the  time  of  the  eclipse  is 
less  than  that  of  the  sun,  (95,)  the  eclipse  cannot  be  total  at  any 
place  ;  but  along  the  line  we  have  described,  the  visible  part  of  the 
sun  will  appear  as  a  very  slender  bright  ring,  encircling  the  moon. 
This  appearance  will  extend  about  fifty  miles  on  each  side,  taking 
in  Burlington,  Middlebury,  Dartmouth,  Bowdoin,  and  Waterville 
colleges  ;  the  ring  will  appear  of  uniform  width  only  along  the 
central  line.     Such  eclipses  as  this  are  called  annular. 


CHAPTER  XIV. 

DELINEATION  OF  A  LUNAR  ECLIPSE. 


125.  The  delineation  of  a  lunar  eclipse  is  extremely  simple, 
since  it  consists  merely  in  representing  the  passage  of  the  moon 
across  the  earth's  shadow.  To  show  the  method  of  effecting  it, 
we  will  proceed  to  delineate  the  lunar  eclipse  of  November,  1844, 
from  the  elements  obtained  in  chapter  11th.     The  angular  dhnen- 

*  By  taking  into  account  several  minute  circumstances  that  we  have  disregarded,  the 
central  track  of  the  eclipse  may  vary  slightly  from  this  description,  probably  passing  ten 
or  fifteen  miles  further  north,  and  nearly  over  Bowdoin  college. 


81 

sions  given  in  the  4th,  6th,  8th  and  10th  elements  being  supposed 
to  be  all  taken  at  the  same  distance,  viz.,  the  distance  from  the  earth 
to  the  moon,  will  serve  as  a  measure  for  their  absolute  dimen- 
sions, in  the  same  manner  as  they  did  in  the  solar  eclipse. 

The  first  step  is  to  make  the  larger  and  smaller  scales  SS  and 
ss  (Fig.  21)  just  as  was  done  in  the  solar  eclipse,  (101  and  109.) 
Take  from  the  longer  one  the  semidiameter  of  the  earth's  shadow, 
viz.,  0°.6306,  and  with  it  describe  the  graduated  circle  BDAE, 
which  will  represent  the  shadow.  It  is  evident  that  the 
plane  of  the  ecliptic  must  bisect  the  shadow,  apd  we  therefore 
draw  the  two  diameters  AB  and  DE  at  right  angles  to  each  other, 
the  former  to  represent  a  section  of  the  plane  of  the  ecliptic,  and 
the  latter  its  axis. 

120.  The  moon's  latitude  is  0°.1810  north.  We  therefore  take 
this  distance  from  the  scale  SS,  and  measure  it  upward  from  C 
toward  D,  which  gives  M  as  the  place  of  the  moon's  centre  at  the 
time  of  full  moon.  If  the  latitude  were  south,  the  centre  would  be 
found  in  the  line  CE. 


127.  Its  path  makes  an  angle  of  5°  45'  41"  with  the  ecliptic, 
tending  south.  If,  therefore,  we  draw  CF,  making  an  angle  of  that 
size  with  CB,  and  YMZ  parallel  to  it,  the  latter  line  will  represent 
the  track  of  the  moon's  centre  across  the  shadow.  It  passes  M  at 
40  minutes  and  14  seconds  past  11  in  the  evening,  by  Greenwich 
time,  anp!  its  position  at  any  other  hour  and  minute  may  be  found 
by  graduating  the  line  YZ,  as  directed  in  article  109th. 

128.  By  taking  the  moon's  semidiameter,  0°.2453,  from  the  scale 
SS,  and  with  it  describing  a  circle  from  any  point  in  its  path  as  a 
centre,  the  position  of  the  entire  disc  will  be  shown,  as  it  must  exist 
at  the  time  indicated  at  its  centre  on  the  path.  It  is  drawn  in  the 
plate  in  five  different  positions  :  1st,  when  it  begins  to  impinge  on 
the  shadow  at  G,  which  must  be  the  commencement  of  the  eclipse  : 
2d,  when  it  just  falls  wholly  within  the  shadow  at  H,  at  which 
time  the  eclipse  must  begin  to  be  total :  3d,  when  its  centre  is  at 
N,  found  by  drawing  CN  perpendicular  to  YZ,  and  thus  (Euc.  3, 
2)  bisecting  the  chord,  which  must  be  the  middle  of  the  eclipse  : 
4th,  when  it  begins  to  leave  the  shadow  at  L,  at  which  time  it 

6 


82 

must  cease  to  be  total,  and  5th,  when  it  entirely  leaves  the  shadow 
at  P,  which  must  be  the  end  of  the  eclipse.  The  respective  cen- 
tres are  at  R,  S,  N,  T  and  V,  and  the  time  may  be  determined 
very  nearly  by  the  drawing. 

129.  More  accurate  results  may,  however,  be  obtained  by  cal- 
culating the  length  of  MR,  MS,  MN,  MT  and  MY,  and  then  find- 
ing by  the  relative  hourly  motion  of  the  moon,  how  long  it  must 
take  it  to  pass  over  them.  In  the  right  angled  triapgle  CNM,  the 
side  CM  and  the  angle  MCN  are  known,  being  our  4th  and  lltfr 
elements,  and  we  can  find  CN  and  MN.  The  sirfe  CN  is  common 
to  the  two  right  angled  triangles  SNC  and  RNC,  and  the  sides  CS 
and  CR  are  also  known,  the  former  being  the  difference,  and  the 
lattjr  the  sum  of  our  8th  and  10th  elements.  Jlpnce  NS  anc(  NR 
can  be  found,  and  likewise  their  equals  NT  and  NV.  Now,  by 
adding  and  subtracting  MN,  which  is  known,  we  shall  have  the 
lines  required. 

130.  The  times  obtained  by  this  process  are  as  follows  :— ~. 

Commencement  of  the  eclipse,  -        ? 

Begins  to  be  total,    - 

Middle,  - 

Ceases  to  be  total, 

End  of  the  eclipse, 

Duration  of  total  obscuration,  - 
Duration  of  the  eclipse,     - 

The  foregoing  is  Greenwich  time,  but  can  readily  be  converted 
into  that  of  any  other  place,  by  allowing  for  the  difference  of  lon- 
gitude. 

If  strict  accuracy  were  aimed  at,  the  elements  should  be  calcu- 
lated at  several  intervals  during  the  eclipse,  as  they  are  liable  to 
vary  considerably. 


h. 

m. 

aec. 

9 

48 

59 

10 

57 

31 

11 

42 

42 

12 

27 

53 

1 

36 

24 

1 

30 

22 

3  47  26 

N 


2 


S 

O 

a 

02 

H 

H 

« 
M 


EXPLANATION  OF  TERMS, 

AS    USED    IN    THIS    WORK 


Ecliptic.     The  apparent  annual  path  of  the  sun  through  the  heavens. 

Nodes.     The  points  where  the  orbit  of  a  planet  intersects  the  plane  of  the  ecliptic. 

Ascending  Node.     That  through  which  the  planet  passes  from  the  south  side  of  the 
ecliptic  to  the  north  side. 

Descending  Node.     That  through  which  it  returns  'to  the  south  side. 

Line  of  the  Nodes.     A  straight  line  connecting  the  two  nodes. 

Equinoctial  Points,  or  Equinoxes.    The  point  where  the  ecliptic  intersects  the  plane 
of  the  equator. 

Vernal  Equinox.     That  through  which  the  sun  apparently  passes  from  the  south  side  of 
the  equator  to  the  north  side. 

Autumnal  Equinox.    That  through  which  it  returns  to  the  south  side. 

Solstitial  Points,  or  Solstices.     Points  in  the  ecliptic  midway  between  the  equi- 
noxes. 

Perigee.     The  point  where  the  sun  or  moon  approaches  nearest  the  earth. 

Apogee.     The  point  where  they  are  most  distant  from  the  earth. 

Apsis.     The  common  name  for  apogee  or  perigee. 

Apsides.     The  plural  of  apsis. 

Line  of  the  Apsides.     A  straight  line  joining  the  two  apsides. 

Conjunction.     In  the  same  direction  as  the  sun. 

Opposition.     In  an  opposite  direction  from  the  sun. 

Stzygy.     The  common  name  for  conjunction  and  opposition. 

Quadrature.     Points  in  the  moon's  orbit  midway  between  the  syzygies. 

Radius  Vector.     A  straight  line  drawn  from  a  revolving  body  to  the  centre  about  which 
it  revolves. 

Latitude  of  a  heavenly  body.     Its  distance  north  or  south  of  the  ecliptic. 

Longitude  of  a  heavenly  body.  Its  distance  eastwardly  from  the  vernal  equinox, 
measured  on  the  ecliptic. 

Right  Ascension.     The  same,  measured  on  the  equator. 

Declination.     The  distance  of  a  heavenly  body,  north  or  south,  from  the  equator. 

Lunation.    The  time  from  one  new  or  full  moon  to  another. 

Parallax.  The  apparent  change  in  the  place  of  a  heavenly  body,  when  viewed  from 
different  points.  It  is  always  equal  to  the  angle  which  a  line  connecting  the 
points  of  observation  would  subtend,  when  viewed  from  the  body. 


Note. — It  is  thought  best  to  omit,  in  the  present  edition  of  this 
work,  a  sequel,  or  second  part,  now  in  manuscript,  and  to  which 
there  have  been  several  references  in  the  foregoing  pages,  ex- 
plaining the  method  of  calculating  most  of  the  lunar  motions  and 
inequalities  directly  from  the  laws  of  elliptical  motion  and  the 
principles  of  gravitation,  without  the  aid  of  tables.  It  may  ap- 
pear hereafter. 


Errata. — Page  15th,  &c.  for  elipse  read  ellipse;  page  27,  near 
the  bottom,  for  syzygyes  read  syzygies. 

Page  32,  Fig.  11,  the  lines  NE  and  EP  should  be  in  the  same 
straight  line. 


I 


ASTRONOMICAL    TABLES 


LIST    OF    TABLES 


1.  Elements  of  Orbits  of  Sun  and  Moon. 

2.  Mean  New  Moons  in  March,  &c. 

3.  Mean  Lunations. 

4.  Mean  Motions  in  hours,  minutes  and  seconds. 
5    Days  of  the  Year  reckoned  from  March. 

6.  Annual  Equation  of  the  Moon's  Perigee. 

7.  Annual  Equation  of  the  Moon's  Node. 

8.  Equation  of  the  Sun's  Centre. 

9.  Equation  of  the  Moon's  Centre. 

10.  Annual  Equation  of  the  Moon's  Longitude. 

11.  Secular  Equation  of  the  Moon's  Longitude. 
12    Variation. 

13.  Evection. 

14.  Annual  Equation  of  Variation. 
Annual  Equation  of  Evection. 
Nodal  Equation  of  Moon's  Longitude. 
Reduction  to  the  Ecliptic. 

18.  Lunar  or  Menstrual  Equation  of  the  Sun's  Longitude. 

19.  Lunar  Nutation  in  Longitude. 
Sun's  Semidiameter  and  Hourly  Motion. 

Moon's  Semidiameter,  Hourly  Motion  and  Equatorial  Parallax. 
Do.  as  affected  by  Evection. 
Obliquity  of  the  Ecliptic  to  the  Equator. 
Moon's  Latitude  in  Eclipses. 

Angle  of  the  visible  path  of  the  Moon  with  the  Ecliptic  in  Eclij 
Sun's  Declination. 

27.  First  Preliminary  Equation. 

28.  Second  Preliminary  Equation. 

29.  Augmentation  of  the  Moon's  Semidiameter. 

30.  To  convert  minutes  into  decimals  of  a  degree. 

31.  To  convert  seconds  into  decimals  of  a  degree. 


15. 
16. 
17. 


TABLE     I. 

Elements  of  Orbits  of  Sun  and  Moon. 


Mean  longitude,  Jan.  1,  1801,            .... 

Sun. 

Moon. 

O         /              // 

280  39  13.17 

118  17'    8.3 

Motion  in  100  years  or  36525  days,  -        -        -        - 

36000  46    0.77 

481267  52  41.6 

Mean  longitude  of  perigee,  Jan.  1,  1801, 

279  31     9.71 

266  10    7.5 

Motion  of  do.  eastward  in  100  vears,        ... 

1  42  56.0 

4069    2  46.6 

Longitude  of  Moon's  JNode,  Jan.  1,  1801, 

13  53  17.7 

Motion  of  do.  westward,  in  100  years,       ... 

1934    9  57.5 

TABLE     II. 

Mean  New  Moon,  SfC.  in  March. 


Mean  New  Moon  in 

Sun's  Mean 

Moon's  Mean 

Sun  and  Moon's 

Longitude  of 

Node. 

Year. 

March. 

Anomaly. 

Anomaly. 

Mean  Longitude. 

1800 

d. 
25 

h. 
0 

m. 

18 

sec. 
49 

83°208 

127.960 

2.7°  39 

28.8207 

1810 

5 

6 

36 

43 

63.167 

63.441 

342.8436 

196.4758 

1820 

14 

1 

38 

40 

72.232 

24.740 

352.0799 

2.5671 

1830 

23 

20 

40 

37 

81.295 

346.038 

1.3163 

168.6535 

1840 

3 

2 

58 

31 

61.254 

281.520 

341.4458 

336.3135 

1841 

22 

0 

31 

8 

79.624 

257.140 

359.8335 

315.9844 

1842 

11 

9 

19 

42 

68.888 

206.943 

349.1144 

297.2190 

1843 

0 

18 

8 

17 

53.152 

156.746 

333.3954 

278.4536 

1844 

18 

15 

40 

54 

76.522 

132.366 

356.7831 

253.1245 

1845 

8 

0 

29 

28 

65.786 

82.169 

346.0640 

23^.3591 

1846 

26 

22 

2 

6 

84.156 

57.789 

4.4517 

219.0300 

1847 

16 

6 

50 

40 

73.420 

7.592 

353.7326 

200.2646 

1848 

4 

15 

39 

15 

62.684 

317.395 

343.0135 

181.4991 

1849 

23 

13 

11 

52 

81.054 

293.015 

1.4012 

161.1702 

1850 

12 

22 

0 

27 

70  318 

242.819 

350.6822 

142.4048 

1851 

2 

6 

49 

1 

59.583 

192.622 

339.9631 

123.6394 

1852 

20 

4 

21 

38 

77.952 

168.242 

353.3508 

103.3103 

1853 

9 

13 

10 

13 

67.217 

118-045 

347.6317 

84.5449 

1854 

28 

10 

42 

50 

85.586 

93.665 

6.0194 

64.2158 

1855 

17 

19 

31 

24 

74.851 

43.468 

355.3003 

45.4504 

1856 

7 

4 

19 

59 

64.115 

353.271 

344.5813 

26.6851 

1857 

25 

1 

52 

36 

82.485 

328.891 

2.9689 

6.3560 

1858 

14 

10 

41 

11 

71.749 

278.694 

352.2499 

347.5906 

1859 

3 

19 

29 

46 

61.013 

228.497 

341.5308 

328.8252 

1860 

21 

17 

2 

24 

79.383 

204.117 

359.9185 

308.4961 

1861 

11 

1 

50 

58 

68.647 

153920 

349.1994 

289.7307 

1862 

0 

10 

39 

33 

57.911 

103  723 

338.4803 

270.9653 

1863 

19 

8 

12 

10 

76.281 

79.343 

356.8680 

250.6362 

1864 

7 

17 

0 

45 

65.545 

29.146 

346.1490 

231.8708 

1865 

26 

14 

33 

22 

83.915 

4.766 

4.5366 

211.5417 

1866 

15 

23 

21 

57 

73.179 

314.569 

353.8176 

192.7763 

1867 

5 

8 

10 

31 

62.443 

264.372 

343.0985 

174.0110 

1868 

23 

5 

43 

9 

80.813 

239.992 

1.4862 

153  6819 

1869 

12 

14 

31 

43 

70.077 

189.795 

350.7671 

134.9165 

1870 

1 

23 

20 

18 

59.34i 

139.599 

340.0481 

116.1511 

1871 

20 

20 

52 

55 

77.711 

115.219 

358.4357 

95  8220 

1872 

9 

5 

41 

29 

66.976 

65.022 

347.7167 

77  0567 

1873 

28 

3 

14 

6 

85.346 

40.642 

6.1043 

56'7275 

1874 

17 

12 

2 

41 

74.610 

350.445 

355.3853 

37*9622 

1875 

6 

20 

51 

15 

63.874 

300.248 

344.6662 

19*1968 

1876 

24 

18 

23 

53 

82.244 

275.868 

3.0539 

358*8677 

1877 

14 

3 

12 

28 

71.508 

225.671 

352.3348 

340  1023 

1878 

3 

12 

1 

3 

60.772 

175.474 

341.6158 

3213370 

1879 

22 

9 

33 

40 

79.142 

151.094 

0.0034 

301  0079 

1880 

10 

18 

22 

15 

68.406 

100.897 

349.2844 

2822425 

1881 

0 

3 

10 

50 

57.670 

50.700 

338.5653 

2634771 

1882 

19 

0 

43 

27 

76.040 

26.320 

356.9530 

243  1480 

1883 

8 

9 

32 

2 

65.304 

336.123 

346.2339 

2243826 

1884 

26 

7 

4 

39 

83.674 

311.743 

4.6216 

2040535 

1885 

15 

15 

53 

14 

72.938 

261.546 

353.9025 

185'2881 

1886 

5 

0 

41 

49 

62.202 

211.349 

343.1835 

1665228 

1887 

23 

22 

14 

26 

80.572 

186.969 

1.5712 

1461937 

1888 
1889 

12 

7 

3 

0 

69.836 

136.772 

350.8521 

127*4283 

1 

15 

51 

35 

59.100 

86.575 

340.1330 

108'6629 

1890 

20 

13 

24 

12 

77.470 

62.196 

353.5207 

88*3337 

1891 

9 

22 

12 

47 

66.734 

11.999 

347.8016 

69*5683 

1892 

27 

19 

45 

24 

85.104 

347.619 

6  1893 

492392 

1893 

17 

4 

33 

59 

74.368 

297  422 

3554703 

30*473;) 

1894 

6 

13 

22 

33 

63.632 

247  225 

344.7512 

11*7085 

1895 
1896 

25 

10 

55 

10 

82.002 

222  845 

3  1389 

351*3794 

13 

19 

43 

45 

71.266 

172  648 

352*4 198 

332.6140 

1897 

3 

4 

32 

19 

60.530 

122  451 

341.7007 

313  8486 

1898 

22 

2 

4 

56 

78.900 

98  071 

0.0884 

2^.5195 

1899 

11 

10 

53 

31 

68.164 

47.874 

349.3694 

274.7542 

1900 

0 

19 

42 

6 

57.428 

357.677 

338.6503 

255.9888 

This  Table  shows  the  time  of  New  Moon  in  March,  of  each  year,  with  the  longitudes,  anom- 
alies, &c.  of  the  Sun  and  Moon  at  that  time,  on  the  supposition  that  all  the  motions  are  per- 
torrned  with  uniform  angular  velocity.  r 


TABLE     III. 

Mean  Lunations. 


No. 

Sun's  Mean 

Moon's  Mean 

Sun  and  Moon's 

Longitude  of 

Lun. 

Mean  Lunations. 

Anomaly. 

Anomaly. 

Mean  Longitude. 

Node. 

d. 

A. 

m. 

sec. 

1 

29 

12 

44 

3 

29.105 

2%.817 

2§.1067 

1.5%38 

2 

59 

1 

28 

6 

58.211 

51634 

58  2135 

3.1275 

3 

88 

14 

12 

9 

87.316 

77.451 

87.3202 

4.6914 

4 

118 

2 

56 

12 

116.421 

103.268 

116.4270 

6.2551 

5 

147 

15 

40 

14 

145.527 

129.085 

145.5/337 
174.6405 

7.8189 

5 

177 

4 

24 

17 

174.632 

154.906 

9.3827 

7 

206 

17 

8 

20 

203.738 

180.718 

203.7472 

109465 

8 

236 

5 

52 

23 

232.843 

206.535 

232.8530 

12.5102 

9 

265 

18 

36 

26 

261.948 

232.352 

261.9607 

14.0740 

10 

295 

7 

20 

29 

291.054 

258.169 

291.0674 

15.6378 

11 

324 

20 

4 

32 

320.159 

283.986 

320.1742 

17.2016 

12 

354 

8 

48 

35 

349.264   j 

309.803 

349.2809 

18.7654 

13 

383 

21 

32 

37 

378.370 

335.620 

378.3877 

20.3291 

h 

14 

18 

22 

1 

14.553 

192.908 

*14.5534 

.7819 

Note. — The  true  quantities  for  one  lunation,  from  which  these  tables  are  calcu- 


lated, are  as  follows,  viz. 

Length  of  a  lunation,      -  29d. 

Sun's  mean  motion  in  Anomaly  in  one  lunation, 
Moon's     do.     ----- 
Sun  and  Moon's  mean  motion  in  Longitude,     do. 
Mean  motion  of  the  Node,    do. 


12A. 


44m.    2.88sec. 

29°.10535764 

25  .81692410 

29  .10674457 

1  .56377989 


TABLE     IV. 


Mean  Motions  of  the  Sun  and  Moon. 

Sun's 

Sun's 

Moon's 

Moon's 

Longitude  of 

Days. 

Anomaly. 

Longitude. 

Anomaly. 

Longitude. 

Node. 

1 

8.9856 

8.98565 

1§.0650 

1§.17640 

.05295 

2 

1.9712 

1.97129 

26.1300 

26  35279 

.10591 

3 

2.9568 

2.95694 

39.1950 

39.52919 

.15886 

4 

3.9424 

3.94259 

52  2600 

52  70559 

.21182 

5 

4.9280 

4.92824 

65.3250 

65.88198 

.26477 

6 

5.9136 

5.91388 

78  3900 

79.05838 

.31773 

7 

6  8992 

6.89953 

91.4549 

92.23477 

.37068 

8 

7.8848 

7.88518 

104.5199 

105.41117 

.42364 

9 

8  8704 

8.87083 

117.5849 

118.58757 

.47659 

Hours. 


bun's 
Anomaly. 


.0411 
.0821 
.1232 
.1643 
.2053 
.2464 
.2875 
.3285 
.3696 


feun's 
Longitude. 


.84107 
.08214 
.12321 
.16427 
.205:34 
.24641 
.28748 
.32855 
.36962 


Moon's 
Anomaly. 


0.5444 
1.0887 
1.6331 
2.1775 
2.7219 
3.2662 
3.8106 
4.3550 
4.8994 


Moon's 
Longitude. 


.5^902 
1.09803 
1.64705 
2.19607 
2.74508 
3.29410 
3.84312 
4  39213 
4.94115 


Longitude  of 
Node. 


.00221 
.00441 
.00662 
.00883 
.01104 
.01324 
.01545 
.01765 
.01986 


For  the  moon  increase  this  by  180°. 


TABLE    NO.    IV CONTINUED. 


Min- 

Sun's Lon. 

Moon's 

Moon's  Lon- 

Lon. of 

Se- 

Sun's Lon. 

Moon's  Lon. 

utes. 

and  Anom. 

Anom. 

gitude. 

Node. 

conds. 

and  Anom. 

and  Anom. 

1 

.0<?068 

.0891 

.00°915 

•06004 

1 

.00001 

.0(?015 

2 

.00137 

.0181 

.01830 

.00007 

2 

.00002 

.00030 

3 

.00205 

.0272 

.02745 

.00011 

3 

.00003 

.00046 

4 

.00274 

.0363 

.03660 

.00015 

4 

.00005 

.00061 

5 

.00342 

.0454 

.04575 

.00018 

5 

.00006 

.00076 

6 

.00411 

.0544 

.05490 

.00022 

6 

.00007 

.00091 

7 

.00479 

.0635 

.06405 

.00026 

7 

.00008 

.00107 

8 

.00548 

.0726 

.07320 

.00029 

18 

.00009 

.00122 

9 

.00616 

.0817 

.08235 

.00033 

9 

.00010 

.00137 

TABLE 

V. 

Bays  of 

the  year  reckoned  from  March. 

Mar. 
1 

April. 

32 

May. 

62 

June. 

July. 

Aug. 
154 

Sept. 
185 

Oct. 

Nov. 

Dec. 
276 

Jan. 

Feb. 
338 

93 

123 

215 

246 

307 

2 

33 

63 

94 

124 

155 

186 

216 

247 

277 

308 

339 

3 

34 

64 

95 

125 

156 

187 

217 

248 

278 

309 

340 

4 

35 

65 

96 

126 

157 

188 

218 

249 

279 

310 

341 

5 

36 

66 

97 

127 

158 

189 

219 

250 

280 

311 

342 

6 

37 

67 

98 

128 

159 

190 

220 

251 

281 

312 

343 

7 

38 

68 

99 

129 

160 

191 

221 

252 

282 

313 

344 

8 

39 

69 

100 

130 

161 

192 

222 

253 

283 

314 

345 

9 

40 

70 

101 

131 

162 

193 

223 

254 

284 

315 

346 

10 

41 

71 

102 

132 

163 

194 

224 

255 

285 

316 

347 

11 

42 

72 

103 

133 

164 

195 

225 

256 

286 

317 

348 

12 

43 

73 

104 

134 

165 

196 

226 

257 

287 

318 

349 

13 

44 

74 

105 

135 

166 

197 

227 

258 

288 

319 

350 

14 

45 

75 

106 

136 

167 

198 

228 

259 

289 

320 

351 

15 

46 

76 

107 

137 

168 

199 

229 

260 

290 

321 

352 

16 

47 

77 

108 

138 

169 

200 

230 

261 

291 

322 

353 

17 

48 

78 

109 

139. 

170 

201 

231 

262 

292 

323 

354 

18 

49 

79 

110 

140 

171 

202 

232 

263 

293 

324 

a55 

19 

50 

80 

111 

141 

172 

203 

233 

264 

294 

325 

356 

20 

51 

81 

112 

142 

173 

204 

234 

265 

295 

326 

357 

21 

52 

82 

113 

143 

174 

205 

235 

266 

296 

327 

358 

22 

53 

83 

114 

144 

175 

206 

236 

267 

297 

328 

359 

23 

54 

84 

115 

145 

176 

207 

237 

268 

298 

329 

360 

24 

55 

85 

116 

146 

177 

208 

238 

269 

299 

330 

361 

25 

56 

86 

117 

147 

178 

209 

239 

270 

300 

331 

362 

26 

57 

87 

118 

148 

179 

210 

240 

271 

301 

332 

363 

27 

58 

88 

119 

149 

180 

211 

241 

272 

302 

333 

364 

28 

59 

89 

120 

150 

181 

212 

242 

273 

303 

334 

365 

29 

60 

90 

121 

151 

182 

213 

243 

274 

304 

335 

366 

30 

61 

91 

122 

152 

183 

214 

244 

275 

305 

336 

31 

92 

153 

184 

245 

306 

337 

Tl 

lis  Tal 

jle  shows  the  mc 

nth  and  day  to  wh 

ich  any  number 

jf  days  i 

n  a  year, 

reck 

aned  f 

rom  t 

he  1st 

of  Ma 

rch,  c< 

jrresp< 

)nds. 

TABLE    VI. 

Annual  Equation  of  the  Moon's  Perigee, 
Argument — Sun's  Anomaly. 


r  Arg. 

0° 

2° 

4° 

6° 

8° 

10° 

Arg. 

~0 

.000 

.013 

.026 

.038 

.051 

.063 

+ 
35 

1 

.063 

.076 

.088 

.101 

.113 

.126 

34 

2 

.126 

•138 

.149 

.161 

.172 

.183 

33 

3 

.183 

.194  % 

.205 

.216 

.226 

.236 

32 

4 

.236 

.245 

.255 

.264 

.273 

.282 

31 

5 

.282 

.290 

.298 

.305 

.312 

.319 

30 

6 

.319 

.325 

.331 

.337 

.342 

.347 

29 

7 

.347 

.351 

.355 

.359 

.362 

.365 

28 

8 

.365 

.367 

.369 

.370 

.371 

.371 

27 

9 

.371 

.371 

.371 

.370 

.369 

.367 

26 

10 

.367 

.365 

.362 

.359 

.355 

.351 

25 

11 

.351 

.347 

.342 

.337 

.331 

.324 

24 

12 

.324 

.318 

.311 

.304 

.296 

.288 

23 

13 

.288 

.280 

.271 

.261 

.252 

.242 

22 

14 

.242 

.231 

.221 

.210 

.199 

.188 

21 

15 

.188 

.177 

.165 

.153 

.141 

.129 

20 

16 

.129 

.117 

.104 

.092 

.079 

.066 

19 

17 

.066 

.052 

.039 

.026 

.013 

.000 

18 

Arg. 

10° 

8° 

6° 

4° 

2° 

0° 

•Arg. 

The  sun's  attraction  causes  a  progressive  motion  in  the  line  of  the  moon's  apsides,  which  affects 
the  place  of  the  perigee;  and  since  the  distance  of  the  sun  varies  in  different  seasons  of  the 
year,  its  attraction  also  varies.  This  causes  the  motion  to  be  more  rapid  at  some  times  than  at 
others,  occasioning  inequalities  in  the  moon's  anomaly,  for  which  this  Table  furnishes  the 
correction. 

TABLE    VII. 

Annual  Equation  of  the  Moon's  Node. 
Argument — Sun's  Anomaly. 


Arg. 

0° 

2° 

4° 

6° 

8° 

10° 

Arg. 

0 

.0000 

.0053 

.0106 

.0159 

.0212 

.0264 

+ 
35 

1 

.0264 

.0316 

.0367 

.0418 

.0469 

.0520 

34 

2 

.0520 

.0570 

.0619 

.0667 

.0714 

.0760 

33 

3 

.0760 

.0805 

.0849 

.0892 

.0934 

.0975 

32 

4 

.0975 

.1015 

.1053 

.1090 

.1126 

.1160 

31 

5 

.1160 

.1192 

.1223 

.1253 

.1281 

.1308 

30 

6 

.1308 

.1333 

.1356 

.1378 

.1398 

.1417 

29 

7 

.1417 

.1434 

.1449 

.1462 

.1373 

.1481 

28 

8 

.1481 

.1488 

.1493 

.1496 

.1499 

.1500 

27 

9 

.1500 

.1498 

.1494 

.1488 

.1481 

.1473 

26 

10 

.1473 

.1463 

.1451 

.1437 

.1421 

.1402 

25 

11 

.1402 

.1381 

.1359 

.1336 

.1313 

.1289 

24 

12 

.1289 

.1263 

.1235 

.1205 

.1273 

.1138 

23 

13 

.1138 

.1103 

.1067 

.1030 

.0992 

.0953 

22 

14 

.0953 

.0913 

.0871 

.0828 

.0784 

.0740 

21 

15 

.0740 

.0695 

.0648 

.0600 

.0551 

.0501 

20 

16 

.0501 

.0452 

.0403 

.0355 

.0306 

.0257 

19 

17 

.0257 

.0206 

.0155 

.0104 

.0052 

2° 

.0000 

18 

Arg. 

10° 

8° 

6° 

4° 

0° 

Arg. 

The  retrograde  motion  of  the  moon's  nodes  is  caused  by  the  sun's  attraction,  and  as  the  dis- 
tance of  that  luminary  varies  in  different  seasons  of  the  year,  its  attraction  must  also  vary,  pro- 
ducing inequalities  in  the  motion  of  the  nodes,  for  which  this  Table  supplies  the  correction. 


TABLE     VIII. 

Equation  of  the  Sun's  Centre. 
Argument — Sun's  Anomaly. 


Arg. 

0° 

1° 

2° 

3o 

40 

50 

6° 

70 

8° 

9° 

10° 

Arg. 
35 

+ 
0 

.0000 

.0343 

.0685 

.1028 

.1370 

.1711 

.2052 

.2393 

.2733 

.3071 

.3409 

1 

.3409 

.3746 

.4081 

.4415 

.4748 

.5079 

.5408 

.5736 

.6062 

•6386 

.6708 

34 

2 

.6708 

.7028 

.7345 

.7660 

.7972 

.8282 

.8590 

.8895 

.9196 

.9495 

.9790 

33 

3 

.9790 

1.0083 

1.0372 

1.0658 

1.0941 

1.1220 

1.1493  1.1767 

1.2035 

1.2299 

1.2559 

32 

4 

1.2559 

1.2815 

1.3067 

1.3315 

1.3559 

1.3799 

1.403411.4264 

1.4491 

1.4711 

1.4928 

31 

5 

1.4928 

1.5140 

1.5347 

1.5549 

1.5747 

1.5939 

1.6127  1.6309 

1.6486 

1.6658 

1.6824 

30 

6 

1.6824 

1.6986 

1.7142 

1.7293 

1.7439 

1.7578 

1.7712  1.7841 

1.7965 

1.8112 

1.8194 

29 

7 

1.8194  1.8301 

1.8401 

1.8496 

1.8585 

1.8669 

1.8747  1.8819 

1.8885 

1.8945 

1.9000 

28 

8 

1.9000  1.9049 

1.9091 

1.9128 

1.9159 

1.9185 

1.9204  1.9217 

1.9225 

1.9226 

1.9223 

27 

9 

1.9223  1.9213 

1.9197 

1.9175 

1.9148 

1.9115 

1.9075  1.9030 

1.8980 

1.8924 

1.8862 

26 

10 

1.8862 

1.8794 

1.8721 

1.8642 

1.8557 

1.8467 

1.8372|  1.8270 

1.8164 

1.8052 

1.7935 

25 

1] 

1.7935 

1.7812 

1.7684 

1.7551 

1.7412 

1.7269 

1.7120;  1.6966 

1.6807 

1.6644 

1.6475 

24 

12 

1.6475 

1.6301 

1.6123 

1.5940 

1.5752 

1.5560 

1.5362J  1.5161 

1.4955 

1.4745 

1.4530 

23 

13 

1.4530 

1.4311 

1.4088 

1.3861 

1.3630 

1.3395 

1.3155  1.2915 

1.2666 

1.2415 

1.2162 

22 

14 

1.2162 

1.1904 

1.1643 

1.1379 

1.1111 

1.0840 

1.0566 

1.0289 

1.0009 

.9726 

.9441 

21 

15 

.9441 

.9152 

.8861 

.8567 

.8271 

.7973 

.7672 

.7369 

.7064 

.6757 

.6448 

20 

16 

.6448 

.6137 

.5825 

.5510 

.5194 

.4877 

.4558 

.4238 

.3917 

.3594 

.3271 

19 

17 

.3271 

.2946 

.2621 

.2296 

.1969 

.1641 

.1314 

.0985 

.0657 

.0329 

.0000 

18 
Arg. 

Arg. 

10o 

9° 

8° 

70 

6° 

50  |  40 

3°   2°  - 

1° 

Oo 

Owing  to  the  elliptical  form  of  the  sun's  apparent  orbit,  it  does  not  revolve  with 
uniform  angular  velocity,  and  this  Table  shows  the  correction  in  its  longitude, 
required  to  be  made  on  that  account. 

The  epoch  of  this  Table  is  1840. 


TABLE     IX. 

Equation  of  the  Moon's  Centre. 
Argument — Moon's  Anomaly. 


Arg. 

+ 
0 

0° 

1° 

2o 

3° 

40 

5° 

6° 

70 

8° 

90 

10O 

Arg. 

0.0000 

0.1181 

0.2362 

0.3542 

0.4721 

0.5897 

0.7072 

0.8244 

0.9412 

1.0578 

1.1738 

35 

1 

1.1738 

1.2896 

1.4048 

1.5223 

1.6336 

1.7471 

1.8600 

1.9721 

2.0835 

2.1944 

2.3041 

34 

2 

2.3041 

2.4131 

2.5212 

2.6284 

2.7346 

2.8398 

2.9439 

3.0470 

3.1490 

3.2498 

3.3494 

33 

3 

3.3494 

3.4478 

3.5450 

3.6409 

3.7354 

3.8287 

3.9205 

4.0109 

4.0999 

4.1874 

4.2735 

32 

4 

4.2735 

4.3580 

4.4410 

4.5223 

4.6022 

4.6804 

4.7569 

4.8318 

4.9050 

4.9765 

5.0463 

31 

5 

5.0463 

5.1143 

5.1806 

5.2451 

5.3078 

5.3687 

5.4277 

5.4849 

5.5403 

5.59385.6454 

30 

6 

5.6454 

5.6952 

5.7480 

5.7889 

5.8330 

5.8750 

5.9152 

5.9534 

5.9897 

6.02406.0564 

29 

7 

6.0564 

6.0868 

6.1152 

6.1417 

6.1663 

6.1889 

6.2094 

6.2281 

6.2448 

6.25956.2722 

28 

8 

6.2722 

6  2830 

6.2919 

6.2988 

6.3033 

6.3068 

6.3079 

6.3070 

6.3043 

6.2997 

6.2931 

27 

9 

6.2931 

6.2847 

6.2743 

6.2621 

6.2481 

6.2322 

6.2144 

6.1948 

6.1734 

6.1502 

6.1252 

26 

10 

6.1252 

6.0984 

6  0699 

6.0396 

6.0076 

5.9739 

5.9384 

5.9013 

5.8624 

5.8220 

5.7799 

25 

11 

5.7799 

5.7362 

5  6908 

5.6439 

5.5954 

5.5453 

5.4937 

5.4406 

5  3860 

5.3300 

5.2723 

24 

12 

5.2723 

5.2135 

5  1531 

5.0913 

5.0281 

4.9636 

4.8978 

4.8306 

4.7622 

4.6924 

4.6215 

23 

13 

4.6215 

4.5493 

4  4759 

4.4013 

4.3256 

4.2487 

4.1707 

4  0915 

4  0114 

3  9302 

3.8480 

22 

14 

3.8480 

3.7620 

3  6806 

3.5954 

3.5093 

3  4223 

3.3344 

3.2457 

3.1562 

3  0658 

2.9747 

21 

15 

2  9747 

2.8828 

2  7901 

2.6968 

2.6028 

2  5081 

2.4127 

2.3168 

2  2203 

2  1232 

2  0256 

20 

16 

2.0256 

1.9275 

18289 

1.7298 

1.6303 

15304 

1.4301 

1.3294 

1.2285 

I  1272 

10256 

19 

17 

Arg. 

1.0256 
~K)o 

0.9238 

0.8217 

0.7194 

0.6170 
6° 

0.5144 

0.4117 

0.3088 

0.2059 

0.1030 
lo 

0.0000 

18 

9° 

8° 

70 

5° 

40 

30 

2o 

00 

Arg. 

Owing  to  the  elliptical  form  of  the  moon's  orbit,  it  does  not  revolve  with  uni- 
form angular  velocity,  and  this  Table  shows  the  correction  in  its  longitude  re- 
quired to  be  made  on  that  account. 

2 


TABLE    X. 

Annual  Equation  of  the  Moorts  Longitude. 
Argument — Sun's  Mean  Anomaly. 


Arg. 
0 

0° 

2° 

4o 

6° 

8° 

10° 
.0318 

Arg. 

+ 
35 

.0000 

.0064 

.0128 

.0192 

.0255 

1 

.0318 

.0381 

.0443 

.0505 

.0567 

.0627 

34 

2 

.0627 

.0687 

.0747 

.0805 

.0863 

.0920 

33 

3 

.0920 

.0974 

.1028 

.1081 

.1132 

.1183 

32 

4 

.1183 

.1232 

.1280 

.1326 

.1370 

.1413 

31 

5 

.1413 

.1454 

.1494 

.1532 

.1568 

.1602 

30 

6 

.1602 

.1634 

.1664 

.1692 

.1719 

.1743 

29 

7 

.1743 

.1764 

.1785 

.1803 

.1818 

.1832 

28 

8 

.1832 

.1843 

.1852 

.1859 

.1864 

.1866 

27 

9 

.1866 

.1865 

.1864 

.1860 

.1853 

.1843 

26 

10 

.1843 

.1832 

.1819 

.1802 

.1784 

.1764 

25 

Jl 

.1764 

.1742 

.1717 

.1690 

.1662 

.1630 

24 

12 

.1630 

.1597 

.1562 

.1525 

.1486 

.1446 

23 

13 

.1446 

.1404 

.1359 

.1313 

.1265 

.1216 

22 

14 

.1216 

.1165 

.1113 

.1059 

.1004 

.0947  • 

21 

15 

.0947 

.0890 

.0831 

.0771 

.0711 

.0649 

20 

16 

.0649 

.05S6 

.0523 

.0459 

.0395 

.0330 

19 

17 

.0330 

.0264 

.0198 

.0132 

.0066 

.0000 

18 

Arg. 

10° 

8° 

6° 

4° 

2° 

0° 

Arg. 

The  attraction  of  the  sun  upon  the  moon  tends  to  draw  it  away  from  the  earth 
and  thus  dilate  its  orbit ;  and  since  the  distance  of  the  sun  from  the  earth  varies 
in  different  seasons  of  the  year,  its  attraction  also  varies.  This  causes  a  variation 
in  the  size  of  the  moon's  orbit,  as  well  as  in  its  velocity,  producing  inequalities  in 
its  longitude,  for  which  this  Table  gives  the  required  corrections. 

TABLE    XI. 

Secular  Equation,  showing  the  Acceleration  of  the  Moon's  Mean 

Motion. 

Argument — The  date. 


Arg. 

0 

5 

181 

00 

01 

182 

01 

02 

183 

03 

04 

184 

05 

06 

185 

07 

09 

186 

11 

13 

187 

15 

17 

188 

19 

21 

189 

24 

27 

190 

30 

33 

The  eccentricity  of  the  earth's  orbit  is,  and  has  been  for  ages,  slowly  diminish- 
ing, which  renders  the  sun's  disturbing  influence  on  the  moon  less  and  less  every 
year,  thus  allowing  the  orbit  of  the  latter  to  contract,  diminishing  its  periodic  time. 
This  Table  contains  the  necessary  corrections  from  this  cause,  at  intervals  of  five 
years  during  the  present  century. 

Note. — The  numbers  in  this  Table  are  the  3d  and  4th  places  of  decimals  of  a 
degree ;  therefore  in  using  them,  two  cyphers  must  be  prefixed  as  decimals. 
Thus  for  33  read  .0033. 


TABLE    XII 
Variation. 


Argument- 

—Moon's  Longitude  diminished 

by  that  of  the  Sun. 

Arg. 

oo 

lo 

2o 

3o 

40     1     50 

60 

70     1     80     1     9° 

lOo 

Arg. 

- 

0 

+.0000 

+.0204 

+.0407 

+.0610 

+.0812+.1013 

+.1212 

+.1410 +.1607 +.1801 

+.1993 

35^ 

i 

+.1993 

+.2183 

+.2370 

+.2553 

+.2734 +.2911 

+.3084 

+.3254 +.3419 +.3580 

+.3736 

34 

2 

+.3736 

+.3887 

+.4034 

+.4175 

+.4312 

+.4443 

+.4568 

+.4687 +.4801  4-4908 

+.5009 

33 

3 

+.5009 

+.5104 

+.5192 

+.5274 

+.5349 

+.5418 

+.5479 

+.5534 +.5581 +.5622 

+.5656 

32 

4 

+.5656 

+.5682 

+.5701 

+.5713 

+.5718 

+.5716 

+.5706 

+.5639 +.5666 +.5634 

+.5597 

31 

■r. 

5 

+.5597 

+.5551 

+.5499 

+.5440  +.5374 

+.5302  +.5222  +.5137|+.5044'+.4946 

+.4841 

30 

6 

+.4841 

+.4730 

+.4613 

+.4490  +.4361 

+.4227 +.4088 +.3944 +.3794 +.3640 

+.3481 

29 

DQ 

7 

+.3481 

+.3317 

+.314S 

+.2978  +.2802 

+.2623  +.2440 +.2254  +.2065  +.1874 

+.1680 

28 

8 

+.1680 

+.1483 

+.1284 

+.1034 

+.0882 

+.0679  +.0475  +.0270  +.0054 

-.0142 

-.0348 

27 

9 

-.0348 

-.0554 

-.0760 

-.0965 

-.1170 

-.1373 

-.1575 

-.1775 

-.1973 

-.2170 

-.2364 

26 

10 

-.2364 

-.2555 

-.2744 

-.2929 

-.3112 

-.3291 

-.3466 

-.3638 

-.3805 

-.3969 

-.4127 

25 

8 

11 

-.4127 

-.4281 

-.4430 

-.4575 

-.4714 

-.4847 

-.4975 

-.5097 

-.5214 

—5324 

-.5429 

24 

- 

12 

-.5429 

-.5526 

-.5618 

-.5703 

-.5781 

-.5853 

-.5918 

-.5976 

-.6026 

-.6070 

-.6107 

23 

l* 

13 

-.6107 

-.6137 

-.6159 

-.6174 

-.6182 

-.6183 

-.6176 

-.6162 

-.6140 

-.6111 

-.6076 

22 

9 

14 

-.6076 

-.6033 

-.5982 

-.5925 

-.5860 

-.5789 

-.5711 

-.5626 

-.5534 

-.5435 

-.5330 

21 

15 

-.5330 

-.5220 

-.5101 

-.4977 

-.4347 

-.4712 

-.4571 

-.4424 

-.4271 

-.4114 

-.3952 

20 

16 

-.3952 

-.3785 

-.3614 

-.3437 

-.3257 

-.3073 

-.2885  -.2694 

-.2500 

-.2302 

-.2102 

19 

17 

-.2102 

-.1899 

-.1694 

-.1487 

-.1278 

-.1067 

-.0855-.0642 

-.0429 

-.0215 

-.0000 
~0o 

18 

- 

Arg. 

10O 

9° 

8o 

70 

60 

50 

40     |    30 

20 

lo 

Arg. 

Reverse  the  Signs. 

Owing  to  the  disturbing  influence  of  the  sun,  the  moon  is  alternately  accelera- 
ted and  retarded  in  the  different  quadrants,  reckoning  from  syzygy.  This  Table 
contains  the  corrections  in  its  longitude  resulting  from  this  cause. 

TABLE    XIII. 
Annual  Equation  of  Variation. 

Arguments — The  argument  of  Variation  at  the  top  and  bottom,  and 
the  Sun's  Mean  Anomaly  at  the  sides. 


130     1    135 

1   270     |    275    |   280    |    285    1    290    |    295    |    300     |   305 

310     1    315 

0    |       5    |      10 

15     1     20    1      25    1      30    1     35 

40 

45 

Arg. 
0 

Arg. 

180    |    185    1    190 

195     I    200    1    205     |    210    |   215 

220 

225 

180 

.00001 -.0064 

-.0125 

-.0LS3 

-.0233 

-.0281 

-.0314 

-.0342 

-.0358 

-.036-1 

180 

360 

10 

190 

+.0008 -.0053 

-.0114 

-.0172 

-.0225 

-.0267 

-.0306 

-.0333 

-.0353 

-.0358 

170 

350 

20 

200 

+.0019  -.0042 

-.0100 

-.0156 

-.0211 

_.0250 

-.0286 

-.0317 

-.0333 

-.0342 

160 

340 

30 

210 

+.0028 -.0028 

-.0083 

-.0133 

-.0181 

-.0225 

-.0261 

-.0286 

-.0306 

-.0314 

150 

330 

40 

220 

+.0033  -.0014 

-.0061 

-.0108 

-.0153 

-.0192 

-.0225 

-.0250 

-.0267 

-.0281 

140 

320 

50 

230 

+.0042+.0000 

-.0039 

-.0081 

-.0119 

-.0153 

-.0181 

-.0206 

-.0225 

-.0233 

130 

310 

60 

240 

+.0047+.0017 

-.0019 

_.0050 

-.0081 

-.0108 

-.0133 

-.0156 

-.0172 

-.0184 

120 

300 

70 

250 

+.0050+.0028 

+.0006 

-.0019 

-.0039 

-.0061 

-.0083 

-.0100 

-.0114 

-.0125 

110 

290 

80 

260 

+.0053  +.0042 

+.0028 

+.0017 

+.0000 

-.0014 

-.0028 

-.0042 

-.0053 

-.0064 

100 

280 

90 

270 

+.0053+.0053 

+.0053 

+.0047 

+.0042 

+.0033 

+.0028 

+.0019 

+.0008 

.0000 

90 

270 

100 

280 

+.0053+.0064 

+.0072 

4..0078 

+.0081 

+.0083 

-I-.0081 

+.0078 

+.0072 

+.0064 

80 

260 

110 

290 

4-.  0050 +.0072 

+.0089 

+  0106 

,.0119 

+.0128 

+.0133  +.0133 

+.0133 

+.0125 

70 

250 

120 

300 

+.0047+.0078 

+.0108 

T.0131 
T.0153 
+.0169 

+.0153 

+.0169 

+.0181 

+.0189 

+.0186 

+.0184 

60 

240 

130 

310 

+.0042+.0031 

+.0119 

■.0181 
T.0206 
T.0225 
+.0233 

+.0206 

+.0206 

+.0233 

+.0239 

+.0233 

50 

230 

140 

320 

+.0033  +.0083 

4-.0128 

+.0236 

+.0225 

+.0275 

+.0281 

+.0281 

40 

220 

150 

330 

+.0028+.0084 

+.0133 

+.0181 

+.0258 

+.0286 

+.0306 

+.0317 

+.0314 

30 

210 

160 

340 

+.0019  4-. 0078 

4-.0133 

+.0189 

+.0275 

+.0306 

+.0328 

+.0339 

+.0342 

20 

200 

170 

350 

+.0003|+.0072 

4-.0133 

+.0186 

+.0239 

+.0281 

+.0317 

+.0339 

+.0353 

+.0358 

10 

190 
Arg. 

90     1     85 

80     1      75 

70 

65    1      60     1      55     1      50     1     45 

Arg. 

270    1   265 

260    1    255 

250 

245    1    240    1    235    I    230     |    225 

180     1    175 

170     1    165     1    160     1    155     1    150     1    145     1    140    1    135 

360    J   355 

350    I    345    1    340    |    335     |    330    |    325    |    320    |    315 

The  inequality  in  the  moon's  motion,  denominated  Variation,  and  for  which 
Table  12th  furnishes  the  correction,  being  occasioned  by  the  disturbing  influence 
of  the  sun,  must  be  greater  or  less  according  as  the  distance  of  the  earth  from  that 
luminary  varies.  In  that  Table  the  earth  is  supposed  to  be  at  its  mean  distance  ; 
hence  another  correction  becomes  necessary,  which  this  Table  furnishes. 


TABLE    XIV. 

Evection. 

Argument — The  Moon's  Mean  Anomaly  diminished  by  twice  the  excess 

of  the  Moon's  Mean  Longitude  over  the  True  Longitude  of  the  Sun. 


Arg. 

0° 

1° 

2° 

3° 

4° 

5° 

6° 

70 

8° 

9° 

10° 

Arg. 

0 

.0000 

.0238 

.0475 

.0713 

.0950 

.1187 

.1423 

.1659 

.1894 

.2129 

.2363 

+ 
35 

1 

.2363 

.2596 

.2829 

.3061 

.3291 

.3521 

.3750 

.3978 

.4204 

.4428 

.4651 

34 

2 

.4651 

.4873 

.5093 

.5312 

.5530 

.5744 

.5958 

.6170 

.6379 

.6587 

.6793 

33 

3 

.6793 

.6996 

.7197 

.7396 

.7593 

.7787 

.7979 

.8168 

.8355 

.8539 

.8720 

32 

4 

.8720 

.8899 

.9074 

.9247 

.9417 

.9584 

.9748 

.9909 

1.0067 

1.0222 

1.0374 

31 

5 

1.0374 

1.0522 

1.0667 

1.0808 

1.0947 

1.1082 

1.1213 

1.1341 

1.1465 

1.1586 

1.1703 

30 

6 

1.1703 

1.1817 

1.1926 

1.2032 

1.2135 

1.2234 

1.2323 

1.2420 

1.2507 

1.2590 

1.2670 

29 

7 

1.2670 

1.2745 

1.2816 

1.2884 

1.2948 

1.3007 

1.3063 

1.3114 

1.3162 

1.3206 

1.3245 

28 

8 

1.3215 

1.3281 

1.3312 

1.3339 

1.3362 

1.3381 

1.3396 

1.3407 

1.3414 

1.3417 

1.3415 

27 

9 

1.3415 

1.3410 

1.3400 

1.3386 

1.3369 

1.3347 

1.3321 

1.3291 

1.3257 

1.3220 

1.3178 

26 

10 

1.3178 

13132 

1.3082 

1.3028 

1.2971 

1.2909 

1.2844 

1.2774 

1.2701 

1.2624 

1.2543 

25 

11 

1.2543 

1.2459 

1.2370 

1.2278 

1.2182 

1.2083 

1.1980 

1.1872 

1.1763 

1.1650 

1.1533 

24 

12 

1.1533 

1.1412 

1.1288 

1.1161 

1.1031 

1.0897 

1.0760 

1.0619 

1.0476 

1.0330 

1.0180 

23 

13 

1.0180 

1.0027 

.9872 

.9713 

.9552 

.9388 

.9221 

.9051 

.8879 

.8704 

.8526 

22 

14 

.8526 

.8346 

.8164 

.7979 

.7792 

.7603 

.7411 

.7217 

.7021 

.6822 

.6622 

21 

15 

.6622 

.6420 

.6217 

.6011 

.5803 

.5594 

.5383 

.5171 

.4957 

.4742 

.4525 

20 

16 

.4525 

.4307 

.4087 

.3867 

.3646 

.3422 

.3199 

.2975 

.2749 

.2523 

.2296 

19 

17 

.2296 
10° 

.2068 

.1840 

.1611 

.1382 

.1152 

.0922 

.0692 

.0462 

.0231 

.0000 

18 

Arg. 

9° 

8° 

7° 

6° 

5° 

40 

3° 

2° 

1<? 

0° 

Arg. 

The  disturbing  influence  of  the  sun  causes  a  variation  in  the  eccentricity  of  the 
moon's  orbit,  and  in  the  position  of  the  major  axis.  This  affects  the  moon's  lon- 
gitude, producing  inequalities,  for  which  this  Table  gives  the  required  corrections. 


TABLE    XV. 

Annual  Equation  of  Evection. 

Arguments — The  argument  for  Evection  at  the  top  and  bottom, 
the  Sun's  Mean  Anomaly  at  the  sides. 


and 


Arg. 

|      0      1      10     1     20     1     30     |     40     i     50      |      60     |      70     1      80    1       90  | 

Arg.  |    180    |    190    1    200    1    210    1    220    1    230    1    240    |     250   1     260    |     270 

0 

180 

.0000i-.0081i-.0l61 

-.0236 

-.0303; -.0364, -.0408 

-.0444, -.0467 

-.0472 

366 

ran 

10 

190 

+.0036 -.0047 -.0122 

-.0200 

-.0272  -.0331 

-.0386 

-.0425! -.0453 

-.0467 

350 

170 

20 

200 

+.0072-00051-0083 

-.0158 

-.0228 

-.0294 

-.0347 

-.0394 

-.0425 

-.0444 

340 

160 

30 

210 

+.0108 +.0033 

-.0042 

-.0111 

-.0181 

-.0244 

-.0303 

-.0347 

-.0386 

-.0408 

330 

150 

40 

220 

+.0136 +.0072 

+.0005 

-.0964 

-.0128 

-.0189 

-.0244 

-.0294 

-.0331 

-.0364 

320 

140 

50 

230 

+.0164+.0108 

+.0050 

-.0011 

-.0072 

-.0128 

-.0181 

-.0228 

-.0272 

-.0303 

310 

130 

60 

240 

+.0186+.0142 

+.0092 

+.0042 

-.0011 

-.0064 

-.0111 

-.0158 

-.0200 

-.0236 

300 

120 

70 

250 

+.0200+.0175+.0133  +.0092 

+.0050 

+.0005 

-.0042 

-.0083 

—0122 

-.0161 

290 

110 

SO 

260 

+.0211  +.01921+.0175+.0142 

+.0108 

+.0072 

+.0033 

-.0005 

-.0047 

-.0081 

280 

100 

90 

270 

+.0211 

+.0211 +.0200 

+.0186 

+.0164 

+.0136 +.0108 

+.0072 

+  0036 

.0000 

270 

90 

100 

280 

+.0211 

+.0219+.0228 

+.0222 

+.0214 

+.0200+.0175 

+.0150 

+.0117 

+.0081 

260 

80 

110 

290 

+.0200 

+.0228+.0242 

+.0256 

+.0258 

+.0253+.0241 

+.0219 

+.0195  +.0161 

250 

70 

120 

300 

+.0186 

+.0222+.0256 

+.0278 

+.0294 

+.0300|+.0297 

+.0286 

+.0264  +.0236 

240 

60 

130 

310 

+.0164 

+.02)4+0258 

+.0294 

+.0320 

+.0339+.0344 

+.0342 

+.0328 

+.0303 

230 

50 

140 

320 

+.0136 

+.0200+.0253 

+.0300 

+.0339 

+.0364+.038:j 

+.0386 

+.0381 

+.0364 

220 

40 

150 

330 

+.0108 

+.0175+.0241 

+.0297 

+.0344 

+.0383+.0405 

+.0422 

,.0422 
+.0450 

+.0408 

210 

30 

160 

340 

+.0072 

+.0150+.0219 

+.0286 

+.0342 

+.0386+.0422 

+.0422 

+.0444 

200 

20 

170 

350 

+.0036 

+.0117  +.0195 

+.0264 

+.0328 

+.0381 +.0422 

+.0450 

+.0464  +.0467 

190 

10 

180 

360 

+.0000 

+.0081 +.0161 

+.02361+.0303 

+.03641+.0408 

+.0444 

+.0467+.0472 

180 

0 

180    i     170                                                                                                  1     90      |Arg. 

Arg. 

360    !    350    i    310    1    330    1    320    1    310    1    300   1     290  |    280    1    270     I 

The  inequality  in  the  moon's  motion  denominated  Evection,  and  for  which  Table  14th  furnishes 
the  correction,  being  occasioned  by  the  disturbing  influence  of  the  sun,  must  be  greater  or  less 
according  as  the  distance  of  the  earth  from  that  luminary  varies.  In  that  Table  the  earth  is 
supposed  to  be  at  its  mean  distance ;  hence  another  correction  becomes  necessary,  which  this 
Table  furnishes. 

Note. — The  foregoing  explanation  is  true  so  far  as  it  goes ;  but  the  values  given  in  the  Table 
are  partially  referrible  to  a  variation  in  the  argument  for  Evection  occasioned  by  the  Annual 
Equation  of  the  Moon's  Longitude.  And  we  might  on  this  principle  add  several  other  Tables, 
which  should  contain  corrections  required  by  the  alteration  of  the  arguments  of  previous  ones; 
but  this  would  not  accord  with  the  simplicity  of  our  design. 


TABLE     XVI. 

Nodal  Equation  of  the  Moon's  Longitude. 
Argument — The  Sun's  longitude  diminished  by  that  of  the  Moon's  Node. 


Arg. 

0° 

2° 

4° 

6" 
.0036 

8° 

10° 

Arg. 

0 

IS 

.0000 

.0012 

.0024 

.0048 

.0059 

+ 
17 

& 

1 

19 

.0059 

.0071 

.0082 

.0092 

.0102 

.0112 

16 

34 

2 

20 

.0112 

.0121 

.0129 

.0137 

.0144 

.0150 

15 

33 

3 

21 

.0150 

.0156 

.0161 

.0165 

.0169 

.0171 

14 

32 

4 

22 

.0171 

.0173 

.0174 

.0174 

.0173 

.0171 

13 

31 

5 

23 

.0171 

.0169 

.0165 

.0161 

.0156 

.0150 

12 

30 

6 

24 

.0150 

.0144 

.0137 

.0129 

.0121 

.0112 

11 

29 

7 

25 

.0112 

.0102 

.0092 

.0082 

.0071 

.0059 

10 

28 

8 

26 

.0059 

.0048 

.0036 

.0024 

4° 

.0012 

~~ 2~ 

.0000 

9 

27 

Arg. 

10° 

8° 

6° 

0° 

Arg. 

The  influence  of  the  sun  tending  to  dilate  the  moon's  orbit  is  greatest  when  the 
former  is  in  the  plane  of  the  latter;  i.e.  when  passing  the  nodes,  and  least  when 
farthest  from  them,  occasioning  an  inequality  in  the  moon's  motion,  for  which 
this  Table  furnishes  the  necessary  correction. 

TABLE     XVII. 

Reduction. 
Argument — The  Longitude  of  the  Moon,  diminished  by  that  of  its  Node. 


Arg. 


o- 


.0000 
.0386 
.0726 
.0979 
.1113 
.1113 
.0979 
.0726 
.0386 


Arg. 


10 


1°  I 

0039 
0423 
0756 
0998 
1119 
1105 
0958 
.0696 
.0349 
~9° 


2° 

0079 
0460 
,0785 
.1016 
1124 


•A 


0937 
.0664 
,0311 


0118 
0496 
.0813 
1033 
1127 
10961.1087 


.0914 
.0632 
0273 

7° 


0157 
0531 
0837 
1048 
1129 
1075 
,0891 
.0599 
.0235 
6° 


J?!!-6.!1 

.0196.0235 

.0565.0599 

.0866  .0891 

.1062 

.1130 

.1062 

.0866 

.0565 

.0196 


9°   10° 


0386 
0726 
0979 
1113 
1113 
0979 
,0726 
.0386 
0000 


OP 


Ar 

*• 

+ 

+ 

17 

35 

16 

34 

15 

33 

14 

32 

13 

31 

12 

30 

11 

29 

10 

28 

9 

27 

Arg. 


The  preceding  Tables  give  the  Moon's  Longitude  in  its  orbit:    this  reduces  it  to 
the  Ecliptic. 

TABLE     XVIII. 

Lunar  or  Menstrual  Equation  of  the  Sun's  Longitude. 
Argument — The  Longitude  of  the  Moon  diminished  by  that  of  the  Sun. 


Arg. 

0° 

5° 

10° 

Arg. 

+ 

— 

+ 

_ 

0 

18 

.0000 

.0002 

.0004 

17 

35 

1 

19 

.0004 

.0005 

.0007 

16 

34 

2 

20 

.0007 

.0009 

.0010 

15 

33 

3 

21 

.0010 

.0012 

.0013 

14 

32 

4 

22 

.0013 

.0015 

.0016 

13 

31 

5 

23 

.0016 

.0017 

.0018 

12 

30 

6 

24 

.0018 

.0019 

.0020 

11 

29 

7 

25 

.0020 

.0020 

.0021 

10 

28 

8 

26 

.0021 

.0021 

.0021 

9 

27 

Arg. 

10° 

5° 

0° 

Arg. 

The  revolution  of  the  earth  around  the  common  centre  of  gravity  of  the  moon 
and  earth,  affects  the  sun's  apparent  place,  causing  a  change  in  its  longitude  for 
which  this  Table  furnishes  the  correction. 


TABLE     XIX. 

Lunar  Nutation  in  Longitude. 
Argument — Longitude  of  the  Moon's  Ascending  Node. 


Arg. 

1     Arg. 

Arg. 

Arg. 

— 

— 

+ 

+ 

0° 

180° 

.0000 

180° 

360° 

10 

170 

.0009 

190 

350 

20 

160 

.0017 

200 

340 

30 

150 

.0024 

210 

330 

40 

140 

.0031 

220 

320 

50 

130 

.0037 

230 

310 

60 

120 

.0042 

240 

300 

70 

110 

.0046 

250 

290 

80 

100 

.0047 

260 

280  ' 

90 

90 

.0048 

270 

270 

The  Precession  of  the  Equinoxes,  being  occasioned  in  part  by  the  attraction  of 
the  moon  lying  out  of  the  plane  of  the  earth's  equator,  must  be  more  or  less  rapid 
according  to  the  obliquity  of  the  plane  of  its  orbit  to  that  of  the  equator,  which 
depends  on  the  longitude  of  tbe  moon's  nodes.  The  longitudes  of  all  the  heav- 
enly bodies,  being  reckoned  from  the  Vernal  Equinox,  must  be  affected  by  any 
change  in  the  place  of  the  point  from  which  they  are  reckoned,  and  therefore  need 
a  correction  from  this  cause,  which  this  Table  supplies. 


TABLE    XX. 

Sun's  Semi-diameter  and  Horary  Motion. 
Argument — Sun's  Anomaly. 


Arg. 

Semi-d  ameter. 

Horary  motion. 

Arg. 

0° 

.2717 

.0425 

360° 

10 

.2716 

.0424 

350 

20 

.2714 

.0423 

340 

30 

.2710 

.0422 

330 

40 

.2705 

.0421 

320 

50 

.2700 

.0419 

310 

60 

.2694 

.0417 

300 

70 

.2687 

.0415 

290 

80 

.2679 

.0413 

280 

90 

.2671 

.0411 

270 

100 

.2663 

.0409 

260  • 

110 

.2655 

.0407 

250 

120 

.2648 

.0405 

240 

130 

.2642 

.0403 

230 

140 

.2637 

.0401 

220 

150 

.2632 

.0400 

210 

160 

.2628 

.0399 

200 

170 

.2626 

.0398 

190 

180 

.2625 

.0397 

180 

The  elliptical  shape  of  the  sun's  apparent  orbit  causes  it  to  vary,  both  as  to  ap- 
parent size  and  velocity.  The  values  of  these  are  given  in  this  Table,  at  intervals 
of  10°  throughout  the  entire  orbit. 


TABLE    XXI. 

Moon's  Semi-diameter,  Horary  Motion  and  Equatorial  Parallax. 
Argument — Moon's  corrected  Anomaly. 


Argument. 

Semi-diam  -ter. 

Horary  motion. 

Equat.  Parallax. 

Argument. 

0° 

.2743 

.6133 

1°.0051 

360 

5 

.2742 

.6131 

1°.0049 

355 

10 

.2740 

.6122 

1°.0042 

350 

15 

.2737 

.6106 

1°.0030 

345 

20 

.2733 

.6086 

1°.0014 

340 

25 

.2727 

.6061 

.9993 

335 

30 

.2719 

.6031 

.9968 

330 

35 

.2712 

.5994 

.9939 

325 

40  • 

.2703 

.5956 

.9907 

*>0 

45 

.2693 

.5911 

.9871 

315 

50 

.2682 

.5864 

.9833 

310 

55 

.2671 

.5816 

.9792 

305 

60 

.2659 

.5767 

.9749 

300 

65 

.2647 

.5714 

.9705 

295 

70 

.2635 

.5661 

.9659 

290 

75 

.2623 

.5608 

.9613 

285 

80 

.2611 

.5556 

.9567 

280 

85 

.2599 

.5503 

.9521 

275 

90 

.2586 

.5450 

.9475 

270 

95 

.2574 

.5397 

-  .9429 

265 

100 

•2562 

.5350 

.9384 

260 

105 

.2550 

.5303 

.9342 

255 

110 

.2539 

.5256 

.9302 

250 

115 

.2528 

.5214 

.9263 

245 

320 

.2518 

.5173 

.9227 

240 

125 

.2509 

.5136 

.9194 

235 

130 

.2500 

.5100 

.9162 

230 

135 

.2492 

.5069 

.9133 

225 

140 

.2485 

.5039 

.9108 

220 

145 

.2479 

.5014 

.9086 

215 

150 

.2474 

.4992 

.9066 

210 

155 

.2469 

.4972 

.9048 

205 

160 

.2465 

.4958 

.9034 

200 

165 

.2462 

.4944 

.9023 

195 

370 

.2460 

.4936 

.9016 

190 

175 

.2459 

.4932 

.9011 

185 

180 

.2458 

.4930 

.9009 

180 

The  principle  and  construction  of  this  Table  is  the  same  as  that  of  Table  20th. 
At  the  time  of  new  or  full  moon  the  quantities  in  this  Table  must  be  increased 
for  the  effect  of  Variation  as  follows,  viz.  1st  column,  .0020;  2d  do.  .0115;  3d  do. 
.0073. 


TABLE     XXII. 

Moon's  Semi-diameter,  Hourly  Motion,  and  Equatorial  Parallax,  as 
affected  by  Evection. 

Argument — The  same  as  for  Evection,  Table  14th. 


Arg. 

Semi-diameter. 

Hourly  motion. 

Equatorial  Parallax. 

Arg. 

0° 

+  .0029 

+.0112 

+.0105 

360° 

10 

-f.0028 

+.0109 

+.0103 

350 

20 

+.0027 

+.0103 

+.0098 

340 

30 

+.0025 

+.0097 

+.0091 

330 

40 

+.0022 

+.0086 

+.0081 

320 

50 

+.0019 

+.0070 

+  .0068 

310 

60 

+.0014 

+  .0055 

+.0052 

300 

70 

+  .0009 

+  .0036 

+.0035 

•   290 

80 

+.0005 

+.0019 

+.0018 

280 

90 

-  .0000 

-  .0002 

-  .0001 

270 

100 

-  .0005 

-  .0020 

-  .0019 

260 

110 

-.0010 

-  .0038 

-  .0037 

250 

120 

-  .0014 

-  .0055 

-  .0054 

240 

130 

-  .0019 

- .0071 

-  .0068 

230 

140 

-.0022 

-.0083 

-  .0080 

220 

150 

- .0024 

-.0094 

-  .0089 

210 

160 

-  .0026 

-  .0103 

-  .0096 

200     . 

170 

-.0027 

-.0107 

-  .0100 

190 

180 

-.0028 

-  .0109 

-  .0103 

180 

All  the  inequalities  in  the  moon's  longitude,  for  which  the  foregoing  Tables 
give  the  corrections,  must  likewise  affect  its  apparent  size,  hourly  motion,  and 
equatorial  parallax.  Variation  and  Evection  are  the  only  ones  that  it  is  important 
to  take  into  account,  the  former  of  which  may  be  considered  constant  at  the  time 
of  new  or  full  moon,  and  this  Table  gives  the  requisite  correction  for  the  latter. 

TABLE    XXIII. 

Obliquity  of  the  Ecliptic  to  the  Equator. 
Argument — The  date. 


Arg. 

184 

185 
186 

18? 

188 
189 


23  27  15.1 
23  27  24  3 


23  27  43.2 
23  27  26.0 


23  27  33.6;23  27  30  7 
23  27  18.4  23  27  20.1 
23  27  20.8:23  2717.4 
23  27  13.4i23  2716.0 


3 


23  27 40.423  27 37. 1  23  27  33.5 
23  27  28.3  23  27  31.0  23  27  33.6 
23  27  27.2  23  27  23.7  23  27  21.3 
23  27  235  23  27  26  0  23  27  27.9 
23  27  13.9  23  27  10.8  23  27  8.4 
23  27  18.3  23  27  20.0  23  27  20.7 


o  /  II 

23  27  29.6 
23  27  35.7 
23  27  17.6 
23  27  29.0 
23  27  6.8 
23  27  20  5 


6 


23  27  27.1 
23  27  37.2 
23  27  15.8 
23  27  29. 
23  27  6.3 
23  27  19.4 


23  2724.8 
23  27  377 
23  2714.9 
27  28.5 
23  27  7.0 
23  2717.3 


2  23 


23  2723.6 
23  27  37.3 
23  27  15.1 
23  2726  7 
23  27  8.6 
23  27  14.4 


23  27  23.4 
23  27  36.0 
23  27  15.9 
23  27  24.1 
23  27  10.8 
23  2711.0 


The  obliquity  of  the  ecliptic  to  the  equator  is  slowly  diminishing,  owing  to  the 
attraction  of  the  planets,  and  is  also  subject  to  an  inequality  whose  period  is  about 
nineteen  years,  caused  by  the  attraction  of  the  moon,  and  called  Nutation.  This 
Table  gives  the  obliquity  on  the  1st  of  January  in  each  year,  taking  both  these 
causes  into  account. 


TABLE    XXIV. 

MoorCs  Latitude  in  Eclipses. 
Argument — Moon's  Longitude  diminished  by  that  of  its  Node. 


Arg. 


S.D. 

lsoio.oboo 

181J0.0873 
182!0.1746 


1880.6950 


0.2617 
0.3486 
0.4353 
0.5219  0 
0.6085 


0.7811 
0.8672 
0.9529 
1.0382 
1.1233 
1.2082 
1.2925 
1.3765 
Arg.      1°.0 


0087 
0960 
1834 
2704 
3572 
4440 
5306 
6173 
7036 
7897 
8758 
9614 
0467 
1318 
2167 
3009 
3849 


0.0175  0 


.9 


0.1048 
0.1921 
0.2791 
0.3659 
0.4526 
0.5392 
0.6258 
0.7122 
0.7983 
0.8844 


0.9700  0 


1.0552 


1.14031 
1.2251  1 
1.3093  1 
1.3933  1 


JB 


0262 

1135 

2008 

287810 

37460 

46130 

54780 

63430 

72080 

80690 

8930;0 

97850 

0638  1 

1488 

2335 

3177 

4017 


.7 


0350 
1222 
2095 
,2965 
3832 
4699 
5565 
6431 
7295 
8155 
9015 
9870 
0723 
1573 
2420 
3261 
4101 


.6 


0.0437 
0.1310 
0.2182 
0.3052 
0.3919 
0.4786 
0.5652 
0.6517 
0.7381 
0.8241 
0.9101 
0.9955 
1.0808 
1.1658 
1.2504 
1.3345 
1.4185 


.5 


0524 
1397 
2269 
3139 
4005 
4873 
5739 
6604 
7467 
8327 
9187 
0041 
0893 
1743 
2588 
3428 
4269 


0.0611 
0.1484 
0.2356 
0.3226 
0.4092 
0.4960 
0.5826 
0.6690 
0.7553 
0.8413 
0.9273 
1.0126 
1.0978 
1.1828 
1.2672 
1.3512 
1.4353 


4  1  3. 


.8 


0.0698 
0.1572 
0.2443 
0.3313 
0.4180 
0.5046 
0.5912 
0.6778 
0.7639 
0.8499 
0.9358 
1.0211 
1.1063 
1.1913 
1.2756 
1.3596 
1.4437 


.2 


.9   lo.O  Arg. 


0.07850 

0.1659 

0.2530 

0.3399 

0.4267 

0.5133 

0.5999 

0.6864 

0.7725 

0.8585 

0.9444 

1.0296 

1.1148 

1  1998  I 

l!2841 1 

1  3680  1 

1.4521 


N.D.  S-A 

73  1791359 
1746  178  358 
2617,177  357 
3486176356 
4353  175,355 
5219!  1741 354 
60851173  353 
6950'172  352 
7811J171  351 
8672170i  350 
9529169  349 
0382|168  348 
1233167,347 
2082166  346 
.2925165  345 
.3765  164  344 


1-4605  163]  343 


Arg. 


The  moon  has  sometimes  a  north  and  sometimes  a  south  latitude,  owing  to  the 
obliquity  of  the  plane  of  its  orbit  to  that  of  the  ecliptic.  This  Table  gives  the 
latitude  for  every  tenth  of  a  degree  of  longitude,  reckoned  17°  either  way  from 
each  node.  The  capital  letters  at  the  hea'd  of  the  columns  of  the  argument  show 
whether  the  latitude  is  north  or  south,  and  whether  it  is  ascending  or  descending. 


TABLE    XXV. 

Angle  of  the  visible  path  of  the  Moon  with  the  Ecliptic  in  Eclipses. 

Arguments — Horary  motion  of  the  Moon  from  the  Sun  at  the  top,  and 
the  Moon's  distance  from  the  Node  at  the  right  and  left. 


N.  A. 

~6° 

S.  D. 

180° 

.44 

.46 

.48 

.50 

.52 

.54 

.56 

.58 

.60 

N.  D. 

S.  A. 

5°47' 

5°46' 

5°45' 

5044/ 

5°43' 

5°42' 

5°41' 

5°40' 

5°39' 

180° 

360° 

3 

183 

5  46 

5  45 

5  44 

5  43 

5  42 

5  41 

5  40 

5  40 

5  39 

177 

357 

6 

186 

5  45 

5  44 

5  43 

5  42 

5  41 

5  40 

5  39 

5  39 

5  38 

174 

354 

9 

189 

5  42 

5  41 

5  40 

5  39 

5  38 

5  38 

5  37 

5  36 

5  35 

171 

351 

12 

192 

5  39 

5  38 

5  37 

5  36 

5  35 

5  34 

5  34 

5  33 

5  32 

168 

348 

15 

195 

5  35 

5  34 

5  33 

5  32 

5  31 

5  30 

5  30 

5  29 

5  28 

165 

345 

The  angle  of  the  moon's  path  with  the  ecliptic,  which  depends  upon  its  dis- 
tance from  the  node,  is  apparently  increased  by  the  earth's  motion  in  the  same 
direction.  This  Table  gives  the  apparent  angle,  taking  both  these  facts  into  con- 
sideration. 


TABLE    XXVI. 

The  Sun's  Declination. 
Argument — Sun's  Longitude. 


/ 


g 


6  2^/?>.? 


Arg. 

Oo 

1° 

2°     j     3o     j     40 

50 

60 

7o 

80 

9° 

lOo 

Arg. 

N. 

0 

s. 

IS 

0.0000 

0°3980 

0°7961    1°1939 

1.5911 

1°9S83 

2°3S50 

2°7809 

3°1761 

3°5703 

3°9639 

s. 
35 

N. 

17 

I 

iy 

3.9639 

4.3564 

4.7477   5.1380 

5.5267 

5.9142 

6.3000 

6.6839 

7.0664 

7.4472 

7.8259 

34 

16 

Si 

20 

7.8259 

8.2025 

8.5767   8.9489 

9.3189 

9.6858 

10.0506 

10.4128 

10.7720 

11.1284 

11.4817 

33 

15 

B 

2  l 

11.4817 

11.8317 

12.1789,12.5225  12.8628 

13.1997 

13.5328 

13.8622 

14.1877 

14.5092 

14.8270 

32 

14 

4 

22 

14.8270 

15.1402 

I5.4494jl5.7539  16.0542 

16.3500 

16.6408 

16.9273 

17.2086 

17.4850 

17.7561 

31 

13 

b 

23 

17.7561 

18.0223 

18.2833.18.5386  18.7886 

19.0331 

19.2717 

19.5042 

19.7314 

19.9528 

20.1681 

30 

12 

6 

24 

20.1681 

20.3770 

20.579520.776120.9664 

21.1500 

21.3267 

21.4975 

21.6614 

21.8186 

21.9689 

29 

11 

V 

25 

21.9689 

221122 

22.2486  22. 378 1I22.5003 

22.6153 

22.7234 

22.8242 

22.9178 

23.0042 

23.0831 

28 

10 

8 

26 

23.0831 

23.1544 

23.2184  23.2750 

23.3242 
6° 

23.3658 
5° 

23.3997 

23.4261 

23.4450 

23.4564 

23.4603 

27 

9 

Arg. 

10o 

90 

8°     )     7° 

40 

3o 

2° 

10 

Oo 

Arg. 

The  plane  of  the  ecliptic  not  coinciding  with  that  of  the  equator,  the  sun  is 
sometimes  north  of  the  equator  and  sometimes  south.  This  is  called  its  declina- 
tion, and  this  Table  shows  its  amount  for  every  degree  of  longitude.  The  epoch 
of  the  Table  is  1840. 


TABLE    XXVII. 

1st  Preliminary  Equation. 

Argument — Moon's  Anomaly. 


0° 

10 

2o 

30 

40 

50 

6° 

70 

8° 

9=> 

10O 

> 
+ 

h.  ra.  s. 

h.  m.  s. 

h.  111.  s. 

h.  m.  s. 

h.  m.  s. 

h.  m.  s. 

h.  m.  s. 

h.  m.  s. 

h.  ra.  s. 

h.  m.  s. 

h.  m.  s. 

0 

0    0    0 

0    9  34 

0  19    8 

0  28  41 

0  38  13 

0  47  44 

0  57  13 

1    6  41 

1  16    7 

125  31 

134  54 

35 

1 

134  54 

144  16 

153  36 

2    2  53 

2  12    8 

2  21  19 

2  30  28 

2  39  34 

2  48  39 

2  57  43  3    6  45 

34 

2 

3    6  45 

3  15  44 

3  24  42 

3  33  38 

3  42  32 

3    123 

4   0    7 

4    8  47 

4  17  25 

4  26    1 

4  34  33 

33 

3 

4  34  33 

4  43    2 

4  51  15 

4  59  42 

5    7  56 

5  16    5 

5  24    9 

5  32    9 

5  40    4 

5  47  54 

5  55  38 

32 

4 

5  55  38 

6    3  16 

6  10  49 

6  18  18 

6  25  40 

6  32  56 

6  40    6 

6  47    6 

6  54   8 

7    1    2 

7    7  50 

31 

5 

7    7  50 

7  14  30 

7  21    2 

7  27  22 

7  33  36 

7  39  46 

7  45  46 

7  5133 

7  57  23 

8    3  12 

8    8  59 

30 

6 

8    8  59 

8  14  33 

8  20  18 

8  25  44 

8  31    0 

8  36    6 

8  41    2 

8  45  48 

8  50  24 

8  54  50 

8  58    6 

29 

7 

8  58    6 

9    3  13 

9   7   9 

9  10  54 

9  14  28 

9  17  51 

9  21    3 

9  24   4 

9  26  54 

9  29  33 

9  32    1 

28 

A 

9  32    1 

9  34  18 

9  36  24 

9  38  19 

9  40    3 

9  4136 

9  42  59 

9  44  11 

9  45  12 

9  46    3 

9  46  44 

27 

9 

9  46  44 

9  47  14 

9  47  33 

9  47  46 

9  47  54 

9  47  49 

9  47  36 

9  47  13 

9  46  38 

9  45  52 

9  44  53 

26 

K) 

9  44  53 

9  43  42 

9  42  21 

9  40  51 

9  39    8 

9  37  14 

9  35  12 

9  32  58 

9  30  32 

9  27  58  9  25  12 

25 

11 

9  25  12 

9  22  14 

9  19    5 

9  15  43 

9  12    9 

9    8  25 

9   4  31 

9    0  25 

8  56  10 

8  5145  8  47    8 

24 

12 

8  47   8 

8  42  18 

8  37  19 

8  32  11 

8  26  53 

8  2124 

8  15  46 

8    9  57 

8    3  56 

7  57  45 

7  5124 

23 

13 

7  5124 

7  44  51 

7  38    9 

7  31  18 

7  24  10 

7  17    9 

7    9  52 

7   2  24 

6  54  46 

6  47   0 

6  39    4 

22 

14 

6  39   4 

6  30  57 

6  22  41 

6  14  19 

6    5  51 

5  57  17 

5  48  37 

5  39  51 

5  30  57 

5  2156 

5  12  48 

21 

If) 

5  12  48 

5    3  33 

4  54  11 

4  44  42 

4  35    6 

4  25  20 

4  15  26 

4    5  26 

3  55  21 

3  45  11 

3  34  58 

20 

16 

3  34  58 

3  24  42 

3  14  24 

3    4    3 

2  53  38 

2  43    9 

2  32  34 

2  21  54 

2  11  10 

2    0  23 

149  33 

19 

17 

< 

149  33 

138  40 

127  44 

1  16  46 

1    5  48 

0  54  50 

0  43  52 

0  32  54 

0  2156 

0  10  58 

000 

18 

< 

10° 

90 

8° 

70 

6« 

50 

40 

3° 

2o 

1° 

0o 

When  the  moon's  anomaly  is  less  than  180°,  it  is  in  advance  of  its  mean  place  at 
time  of  new  or  full  moon  by  reason  of  the  Equation  of  the  Centre,  but  behind 
it  by  Evection,  (Tables  9  and  14,)  yet  on  the  whole  it  is  in  advance;  conse- 
quently it  will  overtake  the  sun  sooner  than  it  would  otherwise  do,  and  something 
must  be  subtracted  from  the  mean  time.  The  contrary  takes  place  when  the  anom- 
aly is  more  than  180°.  This  Table  shows  the  amount  of  time  to  be  added  or  sub- 
tracted from  these  causes. 


TABLE    XXVIII. 


2d  Preliminary  Equation. 
Argument — Sun's  Anomaly. 


OP 


+ 

0 

1 

2 

■3 

4 

5 

6 

7 

8 

9 
10 
II 
12 
13 
11 

152 
16 
1? 


m.  s. 

0  0 
44  28 
27  31 

7  45 
43  57 
14  49 
39  30 
57  27 

7  59 
10  53 

6  10 
54  4 
35  0  3 

9  36 
38  44 

3  12 
24  10 
42  39 


10° 


1Q 
m.  s 

4  29 
48  52 
3141 
1135 
47  18 
17  35 
4140 
53  52 

8  37 
10  45 

5  18 
52  29 
32  45 

6  45 
35  22 
59  26 
20  6 
38  26 


9  0 


go 


m.  s. 

8  56 
53  13 
35  49 

15  20 
50  36 
20  20 
43  45 

0  12 

9  10 
10  33 

422 
50  50 
30  26 

3  51 
3157 
55  37 

16  0 
34  11 


HO 


:p 


13  23 

57  36 

39  56 

19  5 

53  49 

23  0 

45  44 

126 

9  39 

10  16 

3  23 

49  7 

28  3 


40 


ii.  in. 

0  17 

1  1 
141 

2  22 

2  57 

3  25 

3  47 

4  2 
4  10 
4  9 
4  2 
3  47 
3  25 


50 


0  54  2  57 


28  29 
5146 
1153 
28  55 


7« 


2  25 
147 

1  7 
0  25 


h.  m. 
500  22 
56 

1 
47 

0 
35 
38 
35 

4 

55 
18 
1? 
36 
53 
9 
54 
45 
39 


60 


60 


2  26 

3  0 
3  28 

3  49 

4  3 
4  10 
4  9 
4  1 
3  45 
3  23 
2  54 
2  21 
144 
1  3 
0  21 


h.  m. 

0  26 

1  10 
52 

2  30 

3  3 
3  30 

3  51 

4  4 
4  10 
4  8 
3  59 
3  43 
3  20 
2  51 
2  17 
1  40 
0  59 
0  17 


5° 


4° 


70 


h.  m 

0  31  10 

1  14  49 
156  5 

2  33  35 

3  6  10 
3  32  50 

3  52  49 

4  5  37 
4  10  49 
4  8  21 
3  58  27 
3  4123 
3  17  51 
2  48  30 
2  14  14 
1  36  10 
0  55  1 
0  12  51 


30 


9° 


HO 

m.  s.  ft.  in. 
35  36  0  40 
19  5  123 
12  3 
6  2  40 
6311 
3  37 


0 

37 
9 

35  6 
54  26 

6  29 
10  54 

7  41 
57  2 
39  18 
15  9 
45  18 
10  36 
32  12 
51  4 

8  35 


2rj 


3  55 

4  7 
4  10 
4  6 
3  55 
3  37 
3  12 
2  42 
2  6 
128 
0  46 
0  0 


h.  in.  s. 
0  44  28  35 


10O 


10 


27  31 
2  7  45 

2  43  57 

3  14  49 
3  39  30 

3  57  27 

4  7  59 
4  10  53 
4  6  10 
3  54  4 
3  35  0 
3  9  36 
2  38  44 
2  3  12 
124  10 
0  42  39 
0_4_0 

Oo 


When  the  sun's  anomaly  is  less  than  180°,  it  is  before  and  the  moon  behind  the 
mean  place,  by  reason  of  the  Equation  of  the  Centre  (Table  8)  of  the  former, 
and  the  Annual  Equation  of  the  Longitude  (Table  10)  of  the  latter.  For  both  rea- 
sons, then,  the  moon  will  not  overtake  the  sun  so  soon  as  it  would  otherwise  do, 
and  consequently  something  must  be  added  to  the  mean  time  of  New  or  Full  Moon. 
The  contrary  takes  place  when  the  anomaly  is  more  than  180°  ;  and  this  Table 
shows  the  amount  of  time  to  be  added  or  subtracted  from  these  causes. 


TABLE    XXIX. 

Augmentation  of  the  Moon's  Semi- diameter. 

Argument — Distance  of  the  place  (as  projected  on  the  disc)  from  the 

earth's  centre. 


Tables  21  and  22  show  us  the  apparent  semi-diameter  of  the 
moon  as  viewed  from  the  centre  of  the  earth;  but  the  distance 
of  the  moon  from  any  place  on  the  earth's  surface  at  which  it 
is  visible  (save  when  it  is  in  the  horizon)  is  less  than  from  the 
centre,  which  must  cause  it  to  subtend  a  greater  angle.  This 
Table  shows  the  augmentation  in  the  moon's  apparent  semi- 
diameter  from  this  cause. 


Arg. 

+ 

0 

.0045 

10 

.0045 

20 

.0044 

30 

.0043 

40 

.0041 

50 

.0038 

60 

.0035- 

70 

.0031 

80 

.0024 

90 

.0015 

100 

.0000 

TABLE    XXX. 

To  convert  minutes  into  decimals  of  a  degree. 
Argument — The  number  of  minutes. 


Arg. 
0 

0' 

I' 

2' 

3' 

4' 

5' 

6' 

7' 

8' 
.1333 

9' 

.0000 

.0167 

.0333 

.0500 

.0667 

.0833 

.1000 

.1167 

.1500 

1 

.1667 

.1833 

.2000 

.2167 

.2333 

.2500 

.2667 

.2833 

.3000 

.3167 

2 

.3333 

.3500 

.3667 

.3833 

.4000 

.4167 

.4333 

.4500 

.4667 

.4833 

3 

.5000 

.5167 

.5333 

.5500 

.5667 

.5833 

.6000 

.6167 

.6333 

.6500 

4 

.6667 

.6833 

.7000 

.7167 

.7333 

.7500 

.7667 

.7833 

.8000 

.8167 

5 

.8333 

.8500 

.8667 

.8833 

.9000 

.9167 

.9333 

.9500 

.9667 

.9833 

TABLE     XXXI. 

To  convert  seconds  into  decimals  of  a  degree. 
Argument — The  number  of  seconds. 


Arg. 

0" 

1" 

2" 

3" 

4" 

5" 

6" 
.0017 

7" 

8" 

9" 

0 

.0000 

.0003 

.0006 

.0008 

.0011 

.0014 

.0019 

.0022 

.0025 

1 

.0028 

.0031 

.0033 

.0036 

.0039 

.0042 

.0044 

.0047 

.0050 

.0053 

2 

.0056 

.0058 

.0061 

.0064 

.0067 

.0069 

.0072 

.0075 

.0078 

.0081 

o 

.0083 

.0086 

.0089 

.0092 

.0094 

.0097 

.0100 

.0103 

.0106 

.0108 

4 

.0111 

.0114 

.0117 

.0119 

.0122 

.0125 

.0128 

.0131 

.0133 

.0136 

5 

.0139 

.0142 

.0144 

.0147 

.0150 

.0153 

.0156 

.0158 

.0161 

.0164 

QB5HI 

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